Reed–Solomon error correction

15858: 20755: 20285: 15081: 564:). The first element of a CIRC decoder is a relatively weak inner (32,28) Reed–Solomon code, shortened from a (255,251) code with 8-bit symbols. This code can correct up to 2 byte errors per 32-byte block. More importantly, it flags as erasures any uncorrectable blocks, i.e., blocks with more than 2 byte errors. The decoded 28-byte blocks, with erasure indications, are then spread by the deinterleaver to different blocks of the (28,24) outer code. Thanks to the deinterleaving, an erased 28-byte block from the inner code becomes a single erased byte in each of 28 outer code blocks. The outer code easily corrects this, since it can handle up to 4 such erasures per block. 10972: 20298: 19828: 10313: 15853:{\displaystyle {\begin{bmatrix}\Lambda _{3}S_{6}&x^{8}\\\Lambda _{2}S_{6}+\Lambda _{3}S_{5}&x^{7}\\\Lambda _{1}S_{6}+\Lambda _{2}S_{5}+\Lambda _{3}S_{4}&x^{6}\\S_{6}+\Lambda _{1}S_{5}+\Lambda _{2}S_{4}+\Lambda _{3}S_{3}&x^{5}\\S_{5}+\Lambda _{1}S_{4}+\Lambda _{2}S_{3}+\Lambda _{3}S_{2}&x^{4}\\S_{4}+\Lambda _{1}S_{3}+\Lambda _{2}S_{2}+\Lambda _{3}S_{1}&x^{3}\\S_{3}+\Lambda _{1}S_{2}+\Lambda _{2}S_{1}&x^{2}\\S_{2}+\Lambda _{1}S_{1}&x\\S_{1}\end{bmatrix}}={\begin{bmatrix}Q_{2}x^{8}\\Q_{1}x^{7}\\Q_{0}x^{6}\\0\\0\\0\\\Omega _{2}x^{2}\\\Omega _{1}x\\\Omega _{0}\end{bmatrix}}} 20750:{\displaystyle {\begin{bmatrix}001&000&000&000&000&000&000\\000&001&000&000&000&000&000\\000&000&001&000&000&000&000\\000&000&000&001&000&000&000\\000&000&000&000&001&000&000\\000&000&000&000&000&001&000\\000&000&000&000&000&000&001\end{bmatrix}}{\begin{bmatrix}e_{0}\\e_{1}\\q_{0}\\q_{1}\\q_{2}\\q_{3}\\q_{4}\end{bmatrix}}={\begin{bmatrix}006\\924\\006\\007\\009\\916\\003\end{bmatrix}}} 20280:{\displaystyle {\begin{bmatrix}001&000&928&000&000&000&000\\006&006&928&928&928&928&928\\123&246&928&927&925&921&913\\456&439&928&926&920&902&848\\057&228&928&925&913&865&673\\086&430&928&924&904&804&304\\121&726&928&923&893&713&562\end{bmatrix}}{\begin{bmatrix}e_{0}\\e_{1}\\q_{0}\\q_{1}\\q_{2}\\q_{3}\\q_{4}\end{bmatrix}}={\begin{bmatrix}000\\923\\437\\541\\017\\637\\289\end{bmatrix}}} 12568: 9704: 10967:{\displaystyle {\begin{aligned}&Y_{k}X_{k}^{j+\nu }\Lambda (X_{k}^{-1})=0.\\&Y_{k}X_{k}^{j+\nu }\left(1+\Lambda _{1}X_{k}^{-1}+\Lambda _{2}X_{k}^{-2}+\cdots +\Lambda _{\nu }X_{k}^{-\nu }\right)=0.\\&Y_{k}X_{k}^{j+\nu }+\Lambda _{1}Y_{k}X_{k}^{j+\nu }X_{k}^{-1}+\Lambda _{2}Y_{k}X_{k}^{j+\nu }X_{k}^{-2}+\cdots +\Lambda _{\nu }Y_{k}X_{k}^{j+\nu }X_{k}^{-\nu }=0.\\&Y_{k}X_{k}^{j+\nu }+\Lambda _{1}Y_{k}X_{k}^{j+\nu -1}+\Lambda _{2}Y_{k}X_{k}^{j+\nu -2}+\cdots +\Lambda _{\nu }Y_{k}X_{k}^{j}=0.\end{aligned}}} 12214: 3087: 9335: 2735: 19247:

potential polynomials, until a sufficient number of matching polynomials are produced to reasonably eliminate any errors in the received codeword. Once a polynomial is determined, then any errors in the codeword can be corrected, by recalculating the corresponding codeword values. Unfortunately, in all but the simplest of cases, there are too many subsets, so the algorithm is impractical. The number of subsets is the

14983: 445: 11879: 11535: 708: 12563:{\displaystyle {\begin{bmatrix}S_{1}&S_{2}&\cdots &S_{\nu }\\S_{2}&S_{3}&\cdots &S_{\nu +1}\\\vdots &\vdots &\ddots &\vdots \\S_{\nu }&S_{\nu +1}&\cdots &S_{2\nu -1}\end{bmatrix}}{\begin{bmatrix}\Lambda _{\nu }\\\Lambda _{\nu -1}\\\vdots \\\Lambda _{1}\end{bmatrix}}={\begin{bmatrix}-S_{\nu +1}\\-S_{\nu +2}\\\vdots \\-S_{\nu +\nu }\end{bmatrix}}} 17114: 9699:{\displaystyle {\begin{bmatrix}X_{1}^{1}&X_{2}^{1}&\cdots &X_{\nu }^{1}\\X_{1}^{2}&X_{2}^{2}&\cdots &X_{\nu }^{2}\\\vdots &\vdots &\ddots &\vdots \\X_{1}^{n-k}&X_{2}^{n-k}&\cdots &X_{\nu }^{n-k}\\\end{bmatrix}}{\begin{bmatrix}Y_{1}\\Y_{2}\\\vdots \\Y_{\nu }\end{bmatrix}}={\begin{bmatrix}S_{1}\\S_{2}\\\vdots \\S_{n-k}\end{bmatrix}}} 7196:). The "missing" bits in a shortened code need to be filled by either zeros or ones, depending on whether the data is complemented or not. (To put it another way, if the symbols are inverted, then the zero-fill needs to be inverted to a one-fill.) For this reason it is mandatory that the sense of the data (i.e., true or complemented) be resolved before Reed–Solomon decoding. 8871: 415:(1961). By 1963 (or possibly earlier), J. J. Stone (and others) recognized that Reed Solomon codes could use the BCH scheme of using a fixed generator polynomial, making such codes a special class of BCH codes, but Reed Solomon codes based on the original encoding scheme, are not a class of BCH codes, and depending on the set of evaluation points, they are not even 4942: 14629: 11562: 11218: 7181:. This is because it does not matter to the code how many bits in a symbol are in error — if multiple bits in a symbol are corrupted it only counts as a single error. Conversely, if a data stream is not characterized by error bursts or drop-outs but by random single bit errors, a Reed–Solomon code is usually a poor choice compared to a binary code. 13945: 3082:{\displaystyle C(m)=Am={\begin{bmatrix}1&a_{0}&a_{0}^{2}&\dots &a_{0}^{k-1}\\1&a_{1}&a_{1}^{2}&\dots &a_{1}^{k-1}\\\vdots &\vdots &\vdots &\ddots &\vdots \\1&a_{n-1}&a_{n-1}^{2}&\dots &a_{n-1}^{k-1}\end{bmatrix}}{\begin{bmatrix}m_{0}\\m_{1}\\\vdots \\m_{k-1}\end{bmatrix}}} 16835: 386:). The original encoding scheme described in the Reed & Solomon article used a variable polynomial based on the message to be encoded where only a fixed set of values (evaluation points) to be encoded are known to encoder and decoder. The original theoretical decoder generated potential polynomials based on subsets of 1576: 11211: 8597: 14073: 4739: 4704: 7685:

Designers are not required to use the "natural" sizes of Reed–Solomon code blocks. A technique known as "shortening" can produce a smaller code of any desired size from a larger code. For example, the widely used (255,223) code can be converted to a (160,128) code by padding the unused portion of the

8974:

The advantage of looking at the syndromes is that the message polynomial drops out. In other words, the syndromes only relate to the error, and are unaffected by the actual contents of the message being transmitted. If the syndromes are all zero, the algorithm stops here and reports that the message

398:

like scheme based on a fixed polynomial known to both encoder and decoder, but later, practical decoders based on the original scheme were developed, although slower than the BCH schemes. The result of this is that there are two main types of Reed Solomon codes, ones that use the original encoding

19180:

The decoders described in this section use the Reed Solomon original view of a codeword as a sequence of polynomial values where the polynomial is based on the message to be encoded. The same set of fixed values are used by the encoder and decoder, and the decoder recovers the encoding polynomial

19246:

and which encoding method was used to generate the codeword's sequence of values. The original message, the polynomial, and any errors are unknown. A decoding procedure could use a method like Lagrange interpolation on various subsets of n codeword values taken k at a time to repeatedly produce

13793: 567:

The result is a CIRC that can completely correct error bursts up to 4000 bits, or about 2.5 mm on the disc surface. This code is so strong that most CD playback errors are almost certainly caused by tracking errors that cause the laser to jump track, not by uncorrectable error bursts.

6509: 6661: 14978:{\displaystyle {\begin{aligned}S(x)&=S_{t}x^{t-1}+S_{t-1}x^{t-2}+\cdots +S_{2}x+S_{1}\\\Lambda (x)&=\Lambda _{e}x^{e}+\Lambda _{e-1}x^{e-1}+\cdots +\Lambda _{1}x+1\\\Omega (x)&=\Omega _{e}x^{e}+\Omega _{e-1}x^{e-1}+\cdots +\Omega _{1}x+\Omega _{0}\end{aligned}}} 9992: 11874:{\displaystyle \left(\sum _{k=1}^{\nu }Y_{k}X_{k}^{j+\nu }\right)+\Lambda _{1}\left(\sum _{k=1}^{\nu }Y_{k}X_{k}^{j+\nu -1}\right)+\Lambda _{2}\left(\sum _{k=1}^{\nu }Y_{k}X_{k}^{j+\nu -2}\right)+\cdots +\Lambda _{\nu }\left(\sum _{k=1}^{\nu }Y_{k}X_{k}^{j}\right)=0} 11530:{\displaystyle \left(\sum _{k=1}^{\nu }Y_{k}X_{k}^{j+\nu }\right)+\left(\sum _{k=1}^{\nu }\Lambda _{1}Y_{k}X_{k}^{j+\nu -1}\right)+\left(\sum _{k=1}^{\nu }\Lambda _{2}Y_{k}X_{k}^{j+\nu -2}\right)+\cdots +\left(\sum _{k=1}^{\nu }\Lambda _{\nu }Y_{k}X_{k}^{j}\right)=0} 6316: 13651: 1422: 19822: 10987: 7719:

Daniel Gorenstein and Neal Zierler developed a decoder that was described in a MIT Lincoln Laboratory report by Zierler in January 1960 and later in a paper in June 1961. The Gorenstein–Zierler decoder and the related work on BCH codes are described in a book

7203:

or not depends on subtle details of the construction. In the original view of Reed and Solomon, where the codewords are the values of a polynomial, one can choose the sequence of evaluation points in such a way as to make the code cyclic. In particular, if

17301:

The algebraic decoding methods described above are hard-decision methods, which means that for every symbol a hard decision is made about its value. For example, a decoder could associate with each symbol an additional value corresponding to the channel

13956: 22375: 17228:

at MIT published "Improved Decoding of Reed–Solomon and Algebraic-Geometry Codes" introducing an algorithm that allowed for the correction of errors beyond half the minimum distance of the code. It applies to Reed–Solomon codes and more generally to

17109:{\displaystyle {\begin{aligned}E_{0}&=-{\frac {1}{\Lambda _{v}}}(E_{v}+\Lambda _{1}E_{v-1}+\cdots +\Lambda _{v-1}E_{1})\\E_{j}&=-(\Lambda _{1}E_{j-1}+\Lambda _{2}E_{j-2}+\cdots +\Lambda _{v}E_{j-v})&{\text{for }}t<j<n\end{aligned}}} 12019: 12189: 4471: 3951: 3576: 2602: 410:

and Neal Zierler was described in an MIT Lincoln Laboratory report by Zierler in January 1960 and later in a paper in June 1961. The Gorenstein–Zierler decoder and the related work on BCH codes are described in a book Error Correcting Codes by

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presented a polynomial-time soft-decision algebraic list-decoding algorithm for Reed–Solomon codes, which was based upon the work by Sudan and Guruswami. In 2016, Steven J. Franke and Joseph H. Taylor published a novel soft-decision decoder.

4120:

However, Lagrange interpolation performs the same conversion without the constraint on the set of evaluation points or the requirement of an error free set of message values and is used for systematic encoding, and in one of the steps of the

9734:) are already known by some other method (for example, in an FM transmission, the sections where the bitstream was unclear or overcome with interference are probabilistically determinable from frequency analysis). In this scenario, up to 6077: 394:(encoded message length) values of a received message, choosing the most popular polynomial as the correct one, which was impractical for all but the simplest of cases. This was initially resolved by changing the original scheme to a 13321: 8866:{\displaystyle {\begin{aligned}S_{j}&=r(\alpha ^{j})=s(\alpha ^{j})+e(\alpha ^{j})=0+e(\alpha ^{j})\\&=e(\alpha ^{j})\\&=\sum _{k=1}^{\nu }e_{i_{k}}\left(\alpha ^{j}\right)^{i_{k}},\quad j=1,2,\ldots ,n-k\end{aligned}}} 9288: 6334: 13003: 4116: 16705: 14371: 13475: 4937:{\displaystyle \mathbf {C} =\left\{\left(s_{1},s_{2},\dots ,s_{n}\right)\;{\Big |}\;s(a)=\sum _{i=1}^{n}s_{i}a^{i}{\text{ is a polynomial that has at least the roots }}\alpha ^{1},\alpha ^{2},\dots ,\alpha ^{n-k}\right\}.} 12582:

and sets up the linear system for that value. If the equations can be solved (i.e., the matrix determinant is nonzero), then that trial value is the number of errors. If the linear system cannot be solved, then the trial

13150: 8131: 13789: 6514: 10318: 3324: 19375:

was developed as a decoder that is able to recover the original message polynomial as well as an error "locator" polynomial that produces zeroes for the input values that correspond to errors, with time complexity

9838: 14634: 22381: 13940:{\displaystyle {\begin{bmatrix}732&637\\637&762\end{bmatrix}}{\begin{bmatrix}\Lambda _{2}\\\Lambda _{1}\end{bmatrix}}={\begin{bmatrix}-762\\-925\end{bmatrix}}={\begin{bmatrix}167\\004\end{bmatrix}}} 13488: 7689:

The QR code, Ver 3 (29×29) uses interleaved blocks. The message has 26 data bytes and is encoded using two Reed-Solomon code blocks. Each block is a (255,233) Reed Solomon code shortened to a (35,13) code.

19604: 5529: 8482: 7495: 614:

use Reed–Solomon error correction to allow correct reading even if a portion of the bar code is damaged. When the bar code scanner cannot recognize a bar code symbol, it will treat it as an erasure.

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somewhere along the line, the decoders will still operate. The result will be the inversion of the original data. However, the Reed–Solomon code loses its transparency when the code is shortened (

1856: 15066: 8212: 16840: 8602: 11886: 5641: 3686: 12059: 2153: 20958: 19329: 8367: 7898: 5844: 21719: 10156: 8583: 3809: 3434: 2460: 12612:

are the reciprocals of those roots. The order of coefficients of the error location polynomial can be reversed, in which case the roots of that reversed polynomial are the error locators

6238:

erroneous symbols, i.e., it can correct half as many errors as there are redundant symbols added to the block. Sometimes error locations are known in advance (e.g., "side information" in

6716: 6822: 19591: 336:

erasures at locations that are known and provided to the algorithm, or it can detect and correct combinations of errors and erasures. Reed–Solomon codes are also suitable as multiple-

7643: 10206: 5931: 7343: 512:

In 1996, variations of original scheme decoders called list decoders or soft decoders were developed by Madhu Sudan and others, and work continues on these types of decoders – see

7533: 3738: 3427: 2389: 10308: 1944: 21044: 2052: 1571:{\displaystyle \mathbf {C} ={\Bigl \{}\;{\bigl (}p(a_{1}),p(a_{2}),\dots ,p(a_{n}){\bigr )}\;{\Big |}\;p{\text{ is a polynomial over }}F{\text{ of degree }}<k\;{\Bigr \}}\,.} 1340: 6774: 4009: 13154: 10260: 8913: 2455: 1128: 11206:{\displaystyle \sum _{k=1}^{\nu }\left(Y_{k}X_{k}^{j+\nu }+\Lambda _{1}Y_{k}X_{k}^{j+\nu -1}+\Lambda _{2}Y_{k}X_{k}^{j+\nu -2}+\cdots +\Lambda _{\nu }Y_{k}X_{k}^{j}\right)=0} 7705:

The decoders described in this section use the BCH view of a codeword as a sequence of coefficients. They use a fixed generator polynomial known to both encoder and decoder.

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Authors in Andrews et al. (2007), provide simulation results which show that for the same code rate (1/6) turbo codes outperform Reed-Solomon concatenated codes up to 2 dB (

8289: 12835: 7016: 6151: 5009: 1682: 10078: 5897: 1417: 1025: 8969: 7293: 12672: 12054: 10037: 4013: 19361: 17274: 6971: 4247: 2690: 21089: 19410: 16614: 14273: 11557: 5437: 21123: 13326: 12741: 7222: 6236: 4467: 2664: 1750: 20989: 14068:{\displaystyle {\begin{bmatrix}001&000\\000&001\end{bmatrix}}{\begin{bmatrix}\Lambda _{2}\\\Lambda _{1}\end{bmatrix}}={\begin{bmatrix}329\\821\end{bmatrix}}} 7066: 19244: 19217: 16830: 16803: 16761: 16734: 12637: 7379: 7122: 7096: 6912: 5352: 5320: 3804: 3765: 3351: 3207: 3180: 2317: 2180: 1982: 1890: 1196:

There are different encoding procedures for the Reed–Solomon code, and thus, there are different ways to describe the set of all codewords. In the original view of

1186: 13027: 8942: 8028: 5926: 5875: 5761: 5712: 5381: 5293: 5261: 5232: 5203: 5174: 5145: 5116: 5087: 5038: 4447: 4392: 4334: 4305: 4276: 4189: 4160: 14262:

is an alternate iterative procedure for finding the error locator polynomial. During each iteration, it calculates a discrepancy based on a current instance of Λ(

13655: 9758: 6196: 5557: 4732: 4699:{\displaystyle g(x)=\left(x-\alpha ^{i}\right)\left(x-\alpha ^{i+1}\right)\cdots \left(x-\alpha ^{i+n-k-1}\right)=g_{0}+g_{1}x+\cdots +g_{n-k-1}x^{n-k-1}+x^{n-k}} 4418: 4360: 4215: 1708: 1626: 1154: 9784: 1220:. In order to obtain a codeword of the Reed–Solomon code, the message symbols (each within the q-sized alphabet) are treated as the coefficients of a polynomial 19430: 12605:

found in the last step to build the error location polynomial. The roots of the error location polynomial can be found by exhaustive search. The error locators

10228: 9804: 7667: 7419: 7399: 7266: 7246: 7040: 6932: 6885: 6862: 6842: 6310: 6290: 5732: 5401: 5058: 3967:

coefficients, with the constraint that in order for this to work, the set of evaluation points used to encode the message must be a set of increasing powers of

3602: 3371: 3232: 3227: 3141: 3113: 2730: 2710: 2622: 2409: 2337: 2290: 1600: 1282: 1262: 1238: 1218: 1088: 1068: 1048: 990: 967: 944: 7693:

The Delsarte–Goethals–Seidel theorem illustrates an example of an application of shortened Reed–Solomon codes. In parallel to shortening, a technique known as

9169: 672:

codewords. The code rate is generally set to 1/2 unless the channel's erasure likelihood can be adequately modelled and is seen to be less. In conclusion,

362:

view – with BCH view being the most common, as BCH view decoders are faster and require less working storage than original view decoders.

1094:. In the most useful parameterizations of the Reed–Solomon code, the block length is usually some constant multiple of the message length, that is, the 17279:

In 2023, building on three exciting works, coding theorists showed that Reed-Solomon codes defined over random evaluation points can actually achieve

6198:, the measure of redundancy in the block. If the locations of the error symbols are not known in advance, then a Reed–Solomon code can correct up to 22087: 21777: 1984:, and thus every message can be uniquely mapped to such a polynomial, there are different ways of doing this encoding. The original construction of 14595:

Another iterative method for calculating both the error locator polynomial and the error value polynomial is based on Sugiyama's adaptation of the

7686:

source block with 95 binary zeroes and not transmitting them. At the decoder, the same portion of the block is loaded locally with binary zeroes.

5442: 9814:

There is a linear recurrence relation that gives rise to a system of linear equations. Solving those equations identifies those error locations

6504:{\displaystyle P_{b}\approx {\frac {2^{m-1}}{2^{m}-1}}{\frac {1}{n}}\sum _{\ell =t+1}^{n}\ell {n \choose \ell }P_{s}^{\ell }(1-P_{s})^{n-\ell }} 19192:

described a theoretical decoder that corrected errors by finding the most popular message polynomial. The decoder only knows the set of values

8400: 9006: 7921: 4955:, in which each codeword contains the message as a prefix, and simply appends error correcting symbols as a suffix. Here, instead of sending 2185: 9091: 6253:) is able to correct twice as many erasures as errors, and any combination of errors and erasures can be corrected as long as the relation 692: 14990: 8136: 16591:) using discrete Fourier transform. Since the calculation for a discrete Fourier transform is the same as the calculation for syndromes, 514: 9806:

used thus far. This is why twice as many error correcting symbols need to be added as can be corrected without knowing their locations.

22468: 21841:

D. Gorenstein and N. Zierler, "A class of cyclic linear error-correcting codes in p^m symbols," J. SIAM, vol. 9, pp. 207–214, June 1961

22463: 8293: 7824: 5766: 22116: 19432:

is the number of values in a message. The recovered polynomial is then used to recover (recalculate as needed) the original message.

561: 6656:{\displaystyle P_{b}\approx {\frac {1}{m}}{\frac {1}{n}}\sum _{\ell =t+1}^{n}\ell {n \choose \ell }P_{s}^{\ell }(1-P_{s})^{n-\ell }} 740:

tend to produce errors in short bursts. Correcting these burst errors is a job best done by short or simplified Reed–Solomon codes.

734:, a practice that has since become very widespread in deep space and satellite (e.g., direct digital broadcasting) communications. 6153:

The Reed–Solomon code is optimal in the sense that the minimum distance has the maximum value possible for a linear code of size (

22563: 22538: 22354: 22200: 22038: 21874: 1346:, and the sequence of values is the corresponding codeword. Common choices for a set of evaluation points include {0, 1, 2, ..., 21682:

Hagenauer, J.; Offer, E.; Papke, L. (1994). "11. Matching Viterbi Decoders and Reed-Solomon Decoders in a Concatenated System".

9715:, because then the leftmost matrix would be known, and both sides of the equation could be multiplied by its inverse, yielding Y 21814: 9987:{\displaystyle \Lambda (x)=\prod _{k=1}^{\nu }(1-xX_{k})=1+\Lambda _{1}x^{1}+\Lambda _{2}x^{2}+\cdots +\Lambda _{\nu }x^{\nu }} 213: 19522: 7424: 22404: 22320: 22179: 21988: 21941: 21667: 21571: 14100:

with the log of the roots corresponding to the error locations (right to left, location 0 is the last term in the codeword).

16763:

as syndromes (they're the same) and generate the error locator polynomial using the methods from any of the above decoders.

14149: 10161: 7538: 11883:

These summations are now equivalent to the syndrome values, which we know and can substitute in! This therefore reduces to

9325:

were known (see below), then the syndrome equations provide a linear system of equations that can easily be solved for the

7739: 3776: 1755: 680:, meaning that at least half of all the codewords sent must be received in order to reconstruct all of the codewords sent. 538:

Reed–Solomon coding is very widely used in mass storage systems to correct the burst errors associated with media defects.

17293:

errors) over linear size alphabets with high probability. However, this result is combinatorial rather than algorithmic.

12578:

directly but rather searches for it by trying successive values. The decoder first assumes the largest value for a trial

21561: 13646:{\displaystyle S_{1}=r(3^{1})=3\cdot 3^{6}+2\cdot 3^{5}+123\cdot 3^{4}+456\cdot 3^{3}+191\cdot 3^{2}+487\cdot 3+474=732} 7177:

The Reed–Solomon code properties discussed above make them especially well-suited to applications where errors occur in

5562: 3607: 19817:{\displaystyle b_{i}(e_{0}+e_{1}a_{i})-(q_{0}+q_{1}a_{i}+q_{2}a_{i}^{2}+q_{3}a_{i}^{3}+q_{4}a_{i}^{4})=-b_{i}a_{i}^{2}} 3974: 2081: 456: 20887: 22427: 21691: 19254: 17322:, has spurred interest in applying soft-decision decoding to conventional algebraic codes. In 2003, Ralf Koetter and 10083: 7669:. So choosing a sequence of primitive root powers as the evaluation points makes the original view Reed–Solomon code 22519:

FEC library in C by Phil Karn (aka KA9Q) includes Reed–Solomon codec, both arbitrary and optimized (223,255) version

8542: 8235: 21872:

Guruswami, V.; Sudan, M. (September 1999), "Improved decoding of Reed–Solomon codes and algebraic geometry codes",

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In the variant of this algorithm where the locations of the errors are already known (when it is being used as an

6934:-bit value. The sender sends the data points as encoded blocks, and the number of symbols in the encoded block is 6250: 1947: 189: 2062:. The latter encoding procedure, while being slightly less efficient, has the advantage that it gives rise to a 464: 21422: 14382: 14259: 6779: 6666: 6172:

The error-correcting ability of a Reed–Solomon code is determined by its minimum distance, or equivalently, by

1892:

is the rate. This trade-off between the relative distance and the rate is asymptotically optimal since, by the

1200:, every codeword of the Reed–Solomon code is a sequence of function values of a polynomial of degree less than 922:

The Reed–Solomon code is actually a family of codes, where every code is characterised by three parameters: an

431: 169: 12205:

linear equations, not just one. This system of linear equations can therefore be solved for the coefficients Λ

22447: 21427: 19372: 17307: 7608: 6324: 1130:

is some constant, and furthermore, the block length is equal to or one less than the alphabet size, that is,

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elements called symbols. Reed–Solomon codes are able to detect and correct multiple symbol errors. By adding

571:

DVDs use a similar scheme, but with much larger blocks, a (208,192) inner code, and a (182,172) outer code.

21502:

Gorenstein, D.; Zierler, N. (June 1961). "A class of cyclic linear error-correcting codes in p^m symbols".

21442: 14596: 12014:{\displaystyle S_{j+\nu }+\Lambda _{1}S_{j+\nu -1}+\cdots +\Lambda _{\nu -1}S_{j+1}+\Lambda _{\nu }S_{j}=0} 7298: 743:

Modern versions of concatenated Reed–Solomon/Viterbi-decoded convolutional coding were and are used on the

522: 438: 14094:

The coefficients can be reversed to produce roots with positive exponents, but typically this isn't used:

12184:{\displaystyle S_{j}\Lambda _{\nu }+S_{j+1}\Lambda _{\nu -1}+\cdots +S_{j+\nu -1}\Lambda _{1}=-S_{j+\nu }} 7500: 6319:

Theoretical BER performance of the Reed-Solomon code (N=255, K=233, QPSK, AWGN). Step-like characteristic.

3691: 3380: 2342: 21959:"Randomly Punctured Reed-Solomon Codes Achieve the List Decoding Capacity over Polynomial-Size Alphabets" 21773: 10265: 7714: 1903: 21828:. Explains the Delsarte-Goethals-Seidel theorem as used in the context of the error correcting code for 20994: 2011: 1299: 6721: 719:

One significant application of Reed–Solomon coding was to encode the digital pictures sent back by the

480: 282: 206: 10233: 8876: 3946:{\displaystyle C(m)={\begin{bmatrix}p_{m}(a_{0})\\p_{m}(a_{1})\\\cdots \\p_{m}(a_{n-1})\end{bmatrix}}} 3571:{\displaystyle C(m)={\begin{bmatrix}p_{m}(a_{0})\\p_{m}(a_{1})\\\cdots \\p_{m}(a_{n-1})\end{bmatrix}}} 2597:{\displaystyle C(m)={\begin{bmatrix}p_{m}(a_{0})\\p_{m}(a_{1})\\\cdots \\p_{m}(a_{n-1})\end{bmatrix}}} 2414: 1100: 22076: 21654:(1994), "Reed–Solomon Codes and the Compact Disc", in Wicker, Stephen B.; Bhargava, Vijay K. (eds.), 7127: 5646: 312:

check symbols to the data, a Reed–Solomon code can detect (but not correct) any combination of up to

22543: 22052: 21888: 21447: 20873:

In 2002, an improved decoder was developed by Shuhong Gao, based on the extended Euclid algorithm.

17230: 6976: 6099: 4958: 1946:. Being a code that achieves this optimal trade-off, the Reed–Solomon code belongs to the class of 1631: 696: 22036:

Koetter, Ralf; Vardy, Alexander (2003). "Algebraic soft-decision decoding of Reed–Solomon codes".

19363:

code that can correct 3 errors, the naïve theoretical decoder would examine 359 billion subsets.

16444:

A discrete Fourier transform can be used for decoding. To avoid conflict with syndrome names, let

10042: 6111: 5880: 1400: 998: 545:. It was the first use of strong error correction coding in a mass-produced consumer product, and 21862:

Shu Lin and Daniel J. Costello Jr, "Error Control Coding" second edition, pp. 255–262, 1982, 2004

8947: 7271: 7225: 12642: 12026: 10007: 459:. The first commercial application in mass-produced consumer products appeared in 1982 with the 22568: 22047: 21883: 19334: 17315: 17240: 6937: 6328: 6072:{\displaystyle s(x)\equiv p(x)\cdot x^{t}-s_{r}(x)\equiv s_{r}(x)-s_{r}(x)\equiv 0\mod g(x)\,.} 4220: 3374: 2669: 752: 638: 379: 22007:

Randomly punctured Reed--Solomon codes achieve list-decoding capacity over linear-sized fields

21058: 19379: 11542: 9763:

The rest of the algorithm serves to locate the errors, and will require syndrome values up to

5406: 3146: 22394: 22144: 21095: 12726: 7207: 6242: 6201: 4452: 4449:

is defined as the polynomial whose roots are sequential powers of the Galois field primitive

2631: 1717: 230: 199: 22453: 22417: 20964: 7045: 970: 85: 20290: 19248: 19222: 19195: 17225: 16808: 16781: 16739: 16712: 13950: 12615: 7348: 7101: 7075: 6890: 5325: 5298: 4363: 3782: 3743: 3329: 3185: 3158: 2295: 2158: 1960: 1861: 1191: 1159: 923: 488: 437:

In 1975, another improved BCH scheme decoder was developed by Yasuo Sugiyama, based on the

22458: 22125: 8918: 5902: 5851: 5737: 5688: 5357: 5269: 5237: 5208: 5179: 5150: 5121: 5092: 5063: 5014: 4423: 4368: 4310: 4281: 4252: 4165: 4136: 947: 73: 8: 22488: 21918:. STOC 2023. New York, NY, USA: Association for Computing Machinery. pp. 1488–1501. 17318:

belief propagation decoding methods to achieve error-correction performance close to the

13316:{\displaystyle s(x)=p(x)\,x^{t}-s_{r}(x)=3x^{6}+2x^{5}+1x^{4}+382x^{3}+191x^{2}+487x+474} 9737: 6175: 5536: 4951:

The encoding procedure for the BCH view of Reed–Solomon codes can be modified to yield a

4711: 4397: 4339: 4194: 1687: 1605: 1133: 808: 712: 468: 173: 21718:

Andrews, K.S.; Divsalar, D.; Dolinar, S.; Hamkins, J.; Jones, C.R.; Pollara, F. (2007).

12201:

inclusive, and this equivalence is true for any and all such values. Therefore, we have

9766: 1628:

points, this means that any two codewords of the Reed–Solomon code disagree in at least

553:

use similar schemes. In the CD, two layers of Reed–Solomon coding separated by a 28-way

149: 22329: 22213: 22192: 22011: 21966: 21919: 21742: 21519: 19415: 12693:

are known, the error values can be determined. This can be done by direct solution for

10213: 9789: 9283:{\displaystyle (\alpha ^{j})^{i_{k}}=\alpha ^{j*i_{k}}=(\alpha ^{i_{k}})^{j}=X_{k}^{j}} 9088:

Then the syndromes can be written in terms of these error locators and error values as

7725: 7652: 7404: 7384: 7251: 7231: 7185: 7025: 6917: 6870: 6847: 6827: 6295: 6275: 5717: 5386: 5383:

to find the remainder, and then compensating for that remainder by subtracting it. The

5043: 3587: 3356: 3212: 3126: 3098: 3092: 2715: 2695: 2607: 2394: 2322: 2275: 2069: 1585: 1267: 1247: 1223: 1203: 1073: 1053: 1033: 975: 952: 929: 731: 634: 546: 412: 21613: 21596: 17205:. The Reed–Solomon code achieves this bound with equality, and can thus correct up to 12998:{\displaystyle g(x)=(x-3)(x-3^{2})(x-3^{3})(x-3^{4})=x^{4}+809x^{3}+723x^{2}+568x+522} 2066:, that is, the original message is always contained as a subsequence of the codeword. 1711: 114: 97: 22423: 22400: 22316: 22296: 22175: 21984: 21937: 21799: 21697: 21687: 21663: 21577: 21567: 21542: 836: 727: 472: 407: 266: 60: 3779:

is essentially the same as the encoding procedure; it uses the generator polynomial

3120: 617:

Reed–Solomon coding is less common in one-dimensional bar codes, but is used by the

521:

In 2002, another original scheme decoder was developed by Shuhong Gao, based on the

358:

There are two basic types of Reed–Solomon codes – original view and

22363: 22263: 22236: 22217: 22205: 22159: 22057: 21976: 21929: 21893: 21746: 21734: 21608: 21511: 14104: 12705: 6106: 4111:{\displaystyle a_{0},\dots ,a_{n-1}=\{1,\alpha ,\alpha ^{2},\dots ,\alpha ^{n-1}\}} 822: 764: 382:. Their seminal article was titled "Polynomial Codes over Certain Finite Fields". ( 21632: 12574:, but that number has not been determined yet. The PGZ decoder does not determine 1095: 22390: 22333: 22308: 22288: 22251: 22227:(1960), "Encoding and Error Correction Procedures for the Bose-Chaudhuri Codes", 22224: 21980: 17323: 17167: 16700:{\displaystyle R_{j}=E_{j}=S_{j}=r(\alpha ^{j})\qquad {\text{for }}1\leq j\leq t} 14366:{\displaystyle \Delta =S_{i}+\Lambda _{1}\ S_{i-1}+\cdots +\Lambda _{e}\ S_{i-e}} 6246: 6162: 4952: 3581: 3152: 2063: 1893: 744: 737: 720: 467:

Reed–Solomon codes are used. Today, Reed–Solomon codes are widely implemented in

452: 423: 375: 238: 64: 45: 13470:{\displaystyle r(x)=s(x)+e(x)=3x^{6}+2x^{5}+123x^{4}+456x^{3}+191x^{2}+487x+474} 7535:

is also a polynomial of the same degree, this function gives rise to a codeword

5234:. To get a code that is overall systematic, we construct the message polynomial 22448:

Introduction to Reed–Solomon codes: principles, architecture and implementation

22413: 22247: 21958: 21467: 17237:

algorithm) and is based on interpolation and factorization of polynomials over

7694: 7646: 6323:

The theoretical error bound can be described via the following formula for the

2625: 371: 286: 234: 41: 22475: 21738: 15865:

The extended Euclidean algorithm can find a series of polynomials of the form

7674: 7018:

symbols per block. (This is a very popular value because of the prevalence of

661:

symbols of data, being generated, that are then sent over an erasure channel.

241:

in 1960. They have many applications, including consumer technologies such as

22557: 22367: 22240: 22209: 21701: 21546: 17319: 17280: 17234: 13145:{\displaystyle s_{r}(x)=p(x)\,x^{t}{\bmod {g}}(x)=547x^{3}+738x^{2}+442x+455} 8126:{\displaystyle s(x){\bmod {(}}x-\alpha ^{j})=g(x){\bmod {(}}x-\alpha ^{j})=0} 7019: 6867:

For practical uses of Reed–Solomon codes, it is common to use a finite field

4420:

that is known to both the sender and the receiver. The generator polynomial

4133:

In this view, the message is interpreted as the coefficients of a polynomial

756: 748: 637:-RS, can be used to overcome the unreliable nature of data transmission over 630: 22300: 22061: 21933: 21581: 13784:{\displaystyle S_{2}=r(3^{2})=637,\;S_{3}=r(3^{3})=762,\;S_{4}=r(3^{4})=925} 3955:

The inverse Fourier transform could be used to convert an error free set of

22188: 21829: 21432: 17453:% Multiply the information by X^(n-k), or just pad with zeros at the end to 15862:

The middle terms are zero due to the relationship between Λ and syndromes.

12675: 9724: 9166:

This definition of the syndrome values is equivalent to the previous since

4128: 3147:

Systematic encoding procedure: The message as an initial sequence of values

1028: 542: 460: 427: 329: 297: 278: 22454:

A Tutorial on Reed–Solomon Coding for Fault-Tolerance in RAID-like Systems

22077:"Open Source Soft-Decision Decoder for the JT65 (63,12) Reed–Solomon Code" 21911: 18611:% Prepare a linear system to solve the error polynomial and find the error 18551:% basically is nothing that we could do and we return the received message 17534:% Find the Reed-Solomon generating polynomial g(x), by the way this is the 8216:

The transmitted polynomial is corrupted in transit by an error polynomial

7188:, is a transparent code. This means that if the channel symbols have been 3806:

to map a set of evaluation points into the message values as shown above:

3319:{\displaystyle p_{m}(a_{i})=m_{i}{\text{ for all }}i\in \{0,\dots ,k-1\}.} 355:

is up to the designer of the code and may be selected within wide limits.

22492: 22349: 21963:

2023 IEEE 64th Annual Symposium on Foundations of Computer Science (FOCS)

21651: 21437: 19181:(and optionally an error locating polynomial) from the received message. 18149:% Do the euclidean algorithm on the polynomials r0(x) and Syndromes(x) in 17303: 17221: 7670: 7200: 7189: 7178: 7072:

symbols in the block is a design parameter. A commonly used code encodes

6239: 6087: 3116: 1091: 893: 557: 554: 416: 337: 262: 22548: 22275: 1192:

Reed & Solomon's original view: The codeword as a sequence of values

668:

codewords received at the other end is enough to reconstruct all of the

21659: 21523: 19331:, and the number of subsets is infeasible for even modest codes. For a 17311: 14385:

has a detailed description of the procedure. In the following example,

906: 771: 611: 603: 491: 22163: 21897: 17966:% Prepare for the euclidean algorithm (Used to find the error locating 17870:% If all syndromes are zeros (perfectly divisible) there are no errors 16273:

Using the same data as the Peterson–Gorenstein–Zierler example above:

14405:

Using the same data as the Peterson Gorenstein Zierler example above:

8539:

The decoder starts by evaluating the polynomial as received at points

586:(discontinued in 2015) also used Reed–Solomon when breaking up files. 21720:"The development of turbo and LDPC codes for deep-space applications" 10039:. This follows from the above product notation construction since if 7174:

code, and is capable of correcting up to 16 symbol errors per block.

403: 22533: 22267: 21515: 18548:% If no error position is found we return the received work, because 17801:% Find the syndromes (Check if dividing the message by the generator 2070:

Simple encoding procedure: The message as a sequence of coefficients

2058:

by interpolating these values with a polynomial of degree less than

22016: 21971: 21924: 21916:

Proceedings of the 55th Annual ACM Symposium on Theory of Computing

21417: 17197:

was usually understood to limit the error-correction capability to

6166: 1684:

positions. Furthermore, there are two polynomials that do agree in

599: 575: 476: 395: 359: 242: 22539:

Matlab implementation of errors and-erasures Reed–Solomon decoding

22295:, International Symposium on Information Theory, San Remo, Italy, 22005: 17483:% Get the remainder of the division of the extended message by the 296:

Reed–Solomon codes operate on a block of data treated as a set of

22518: 22500: 12594: 5263:

by interpreting the message as the sequence of its coefficients.

5060:

largest monomials are equal to the corresponding coefficients of

770:

These concatenated codes are now being replaced by more powerful

760: 618: 607: 595: 578:

files which are commonly posted accompanying multimedia files on

258: 254: 22528: 21594: 19498:

Errors in transmission might cause this to be received instead.

16548:), and since a discrete Fourier transform is a linear operator, 13323:

Errors in transmission might cause this to be received instead.

2000:, whereas subsequent constructions interpret the message as the 711:

Deep-space concatenated coding system. Notation: RS(255, 223) +

444: 21910:

Brakensiek, Joshua; Gopi, Sivakanth; Makam, Visu (2023-06-02).

21595:

Sugiyama, Y.; Kasahara, M.; Hirasawa, S.; Namekawa, T. (1975).

17341: 17142:) and take the inverse transform (polynomial interpolation) of 12829: 7673:. Reed–Solomon codes in the BCH view are always cyclic because 707: 579: 499: 290: 21800:"Kissing Numbers, Sphere Packings, and Some Unexpected Proofs" 17306:'s confidence in the correctness of the symbol. The advent of 7098:

eight-bit data symbols plus 32 eight-bit parity symbols in an

5685:

are zero. Therefore, the following definition of the codeword

3377:. Once it has been found, it is evaluated at the other points 1953:

While the number of different polynomials of degree less than

560:

yields a scheme called Cross-Interleaved Reed–Solomon Coding (

347:

consecutive bit errors can affect at most two symbols of size

22256:

Journal of the Society for Industrial and Applied Mathematics

22004:

Alrabiah, Omar; Guruswami, Venkatesan; Li, Ray (2023-08-18),

21717: 18455:% Find the zeros of the error polynomial on this galois field 18314:% Check if the degree of remainder(x) is less than max_errors 17161: 13077: 8089: 8045: 6914:

elements. In this case, each symbol can be represented as an

5497: 3770: 1419:

of codewords of the Reed–Solomon code is defined as follows:

688: 583: 484: 274: 21597:"A method for solving key equation for decoding Goppa codes" 21157:

001 x + 908 x + 175 x + 194 x + 695 x + 094 x + 720 x + 000

17569:% A multiplication on the galois field is just a convolution 12587:

is reduced by one and the next smaller system is examined. (

22529:

Henry Minsky's RSCode library, Reed–Solomon encoder/decoder

22481: 21912:"Generic Reed-Solomon Codes Achieve List-Decoding Capacity" 17217:

errors. However, this error-correction bound is not exact.

16439: 12750:. This is generally done using a precomputed lookup table. 9311:

unknowns, but that system of equations is nonlinear in the

5524:{\displaystyle s_{r}(x)=p(x)\cdot x^{t}\ {\bmod {\ }}g(x).} 3151:

There is an alternative encoding procedure that produces a

684: 21965:. FOCS 2023, Santa Cruz, CA, USA, 2023. pp. 164–176. 20881:

Using the same data as the Berlekamp Welch example above:

17366:% RSENCODER Encode message with the Reed-Solomon algorithm 12832:

barcodes) for a RS(7,3) code. The generator polynomial is

8585:. We call the results of that evaluation the "syndromes", 6973:. Thus a Reed–Solomon code operating on 8-bit symbols has 6086:

The Reed–Solomon code is a code; in other words, it is a

5899:, that is, they are divisible by the generator polynomial 22523: 22315:(Revised ed.), Laguna Hills, CA: Aegean Park Press, 19440:

Using RS(7,3), GF(929), and the set of evaluation points

12570:

The above assumes the decoder knows the number of errors

8486:

The goal of the decoder is to find the number of errors (

8477:{\displaystyle e(x)=\sum _{k=1}^{\nu }e_{i_{k}}x^{i_{k}}} 6315: 550: 422:

In 1969, an improved BCH scheme decoder was developed by

270: 250: 17372:% prim_poly: Primitive polynomial p(x). Ie for DM is 301 14097:

R(x) = 001 x + 821 x + 329, with roots 27 = 3 and 81 = 3

7490:{\displaystyle (p(\alpha ^{1}),\dots ,p(\alpha ^{q-1}))} 4129:

The BCH view: The codeword as a sequence of coefficients

3209:

is defined as the unique polynomial of degree less than

22254:(1960), "Polynomial Codes over Certain Finite Fields", 21798:

Pfender, Florian; Ziegler, Günter M. (September 2004),

17510:% Now return the message with the redundant information 17233:. This algorithm produces a list of codewords (it is a 9318:

and does not have an obvious solution. However, if the

9079:{\displaystyle X_{k}=\alpha ^{i_{k}},\ Y_{k}=e_{i_{k}}} 7995:{\displaystyle g(x)=\prod _{j=1}^{n-k}(x-\alpha ^{j}),} 4875: is a polynomial that has at least the roots 3155:

Reed–Solomon code. Here, we use a different polynomial

2265:{\displaystyle p_{m}(a)=\sum _{i=0}^{k-1}m_{i}a^{i}\,.} 1957:

and the number of different messages are both equal to

402:

Also in 1960, a practical fixed polynomial decoder for

246: 21774:"Analytical Expressions Used in bercoding and BERTool" 20691: 20578: 20307: 20221: 20108: 19837: 19257: 19175: 18929:% Now to fix the errors just add with the encoded code 15695: 15090: 14231:{\displaystyle e_{1}x^{3}+e_{2}x^{4}=74x^{3}+122x^{4}} 14044: 14001: 13965: 13916: 13881: 13838: 13802: 12477: 12407: 12223: 9634: 9570: 9344: 9157:{\displaystyle S_{j}=\sum _{k=1}^{\nu }Y_{k}X_{k}^{j}} 7708: 7598:{\displaystyle (p(\alpha ^{2}),\dots ,p(\alpha ^{q}))} 6669: 6114: 3833: 3458: 3017: 2768: 2484: 629:

Specialized forms of Reed–Solomon codes, specifically

21098: 21061: 20997: 20967: 20890: 20301: 19831: 19607: 19525: 19418: 19382: 19337: 19225: 19198: 17243: 16838: 16811: 16784: 16742: 16715: 16617: 15084: 14993: 14632: 14276: 14152: 13959: 13796: 13658: 13491: 13329: 13157: 13030: 12838: 12729: 12645: 12618: 12217: 12062: 12029: 11889: 11565: 11545: 11221: 10990: 10316: 10268: 10236: 10216: 10164: 10086: 10045: 10010: 9841: 9792: 9769: 9740: 9338: 9172: 9094: 9009: 8950: 8921: 8879: 8600: 8545: 8403: 8296: 8238: 8139: 8031: 7924: 7827: 7806:{\displaystyle (c_{0},\ldots ,c_{i},\ldots ,c_{n-1})} 7742: 7655: 7611: 7541: 7503: 7427: 7407: 7387: 7351: 7301: 7274: 7254: 7234: 7210: 7130: 7104: 7078: 7048: 7028: 6979: 6940: 6920: 6893: 6873: 6850: 6830: 6782: 6724: 6517: 6337: 6298: 6278: 6204: 6178: 5934: 5905: 5883: 5854: 5769: 5740: 5720: 5691: 5649: 5565: 5539: 5445: 5409: 5403:

check symbols are created by computing the remainder

5389: 5360: 5328: 5301: 5272: 5240: 5211: 5182: 5153: 5124: 5095: 5066: 5046: 5017: 4961: 4742: 4714: 4474: 4455: 4426: 4400: 4371: 4342: 4313: 4307:

is constructed by multiplying the message polynomial

4284: 4255: 4223: 4197: 4168: 4139: 4016: 3977: 3812: 3785: 3746: 3694: 3610: 3590: 3437: 3383: 3359: 3332: 3235: 3215: 3188: 3161: 3129: 3101: 2738: 2718: 2698: 2672: 2634: 2610: 2463: 2417: 2397: 2345: 2325: 2298: 2278: 2188: 2161: 2084: 2014: 1963: 1906: 1864: 1851:{\displaystyle \delta =d/n=1-k/n+1/n=1-R+1/n\sim 1-R} 1758: 1720: 1690: 1634: 1608: 1588: 1425: 1403: 1302: 1270: 1250: 1226: 1206: 1162: 1136: 1103: 1076: 1056: 1036: 1001: 978: 955: 932: 21168:

055 x + 440 x + 497 x + 904 x + 424 x + 472 x + 001

16021:) don't need to be saved, so the algorithm becomes: 15061:{\displaystyle \Lambda (x)S(x)=Q(x)x^{t}+\Omega (x)} 14381:

so that a recalculated Δ would be zero. The article

13024:, then a systematic codeword is encoded as follows. 12701: 8207:{\displaystyle s(\alpha ^{j})=0,\ j=1,2,\ldots ,n-k} 7902:

As a result of the Reed-Solomon encoding procedure,

7697:

allows omitting some of the encoded parity symbols.

5011:, the encoder constructs the transmitted polynomial 3963:

message values back into the encoding polynomial of

1027:. The set of alphabet symbols is interpreted as the 475:

standards, though they are being slowly replaced by

451:

In 1977, Reed–Solomon codes were implemented in the

22172:

Error Control Coding: Fundamentals and Applications

22003: 21909: 21681: 17384:% enc_msg = rsEncoder(msg, 8, 301, 12, numel(msg)); 16832:, the error locator polynomial, and these formulas 8378:will be zero if there is no error at that power of 399:scheme, and ones that use the BCH encoding scheme. 22549:Pure-Python implementation of a Reed–Solomon codec 22534:Open Source C++ Reed–Solomon Soft Decoding library 21117: 21083: 21038: 20983: 20952: 20749: 20279: 19816: 19585: 19424: 19404: 19355: 19323: 19238: 19211: 17268: 17108: 16824: 16797: 16755: 16728: 16699: 15852: 15060: 14977: 14365: 14230: 14067: 13939: 13783: 13645: 13469: 13315: 13144: 12997: 12735: 12666: 12631: 12562: 12183: 12048: 12013: 11873: 11551: 11529: 11205: 10966: 10302: 10254: 10222: 10200: 10150: 10072: 10031: 9986: 9798: 9778: 9752: 9698: 9282: 9156: 9078: 8978: 8963: 8936: 8907: 8865: 8577: 8476: 8361: 8283: 8206: 8125: 7994: 7892: 7805: 7661: 7637: 7597: 7527: 7489: 7413: 7393: 7373: 7337: 7287: 7260: 7240: 7216: 7166: 7116: 7090: 7060: 7034: 7010: 6965: 6926: 6906: 6879: 6856: 6844:is the symbol error rate in uncoded AWGN case and 6836: 6816: 6768: 6710: 6655: 6503: 6304: 6284: 6230: 6190: 6145: 6071: 5920: 5891: 5869: 5838: 5755: 5734:coefficients are identical to the coefficients of 5726: 5706: 5677: 5636:{\displaystyle x^{t-1},x^{t-2},\dots ,x^{1},x^{0}} 5635: 5551: 5523: 5431: 5395: 5375: 5346: 5314: 5287: 5266:Formally, the construction is done by multiplying 5255: 5226: 5197: 5168: 5139: 5110: 5081: 5052: 5032: 5003: 4936: 4726: 4698: 4461: 4441: 4412: 4386: 4354: 4328: 4299: 4270: 4241: 4209: 4183: 4154: 4110: 4003: 3945: 3798: 3759: 3732: 3681:{\displaystyle p_{m}(a_{0}),\dots ,p_{m}(a_{k-1})} 3680: 3596: 3570: 3421: 3365: 3345: 3318: 3221: 3201: 3174: 3135: 3107: 3081: 2724: 2704: 2684: 2658: 2616: 2596: 2449: 2403: 2383: 2331: 2311: 2284: 2264: 2174: 2147: 2046: 1976: 1938: 1884: 1850: 1744: 1702: 1676: 1620: 1594: 1570: 1411: 1334: 1276: 1256: 1232: 1212: 1180: 1148: 1122: 1082: 1062: 1042: 1019: 984: 961: 938: 479:. For example, Reed–Solomon codes are used in the 22388: 22380:, MIT Lecture Notes 6.451 (Video), archived from 21501: 17663:% RSDECODER Decode a Reed-Solomon encoded message 17420:% Get the Reed-Solomon generating polynomial g(x) 6597: 6584: 6445: 6432: 4811: 2457:for the Reed–Solomon code is defined as follows: 2148:{\displaystyle m=(m_{0},\dots ,m_{k-1})\in F^{k}} 1559: 1528: 1436: 505:In 1986, an original scheme decoder known as the 22555: 22477:Tutorial on Reed–Solomon Error Correction Coding 22143:Hong, Jonathan; Vetterli, Martin (August 1995), 20953:{\displaystyle R_{-1}=\prod _{i=1}^{n}(x-a_{i})} 19324:{\textstyle {\binom {n}{k}}={n! \over (n-k)!k!}} 18368:% Update r0, r1, g0, g1 and remove leading zeros 12807:Consider the Reed–Solomon code defined in 12711: 10158:, making the whole polynomial evaluate to zero. 8362:{\displaystyle e(x)=\sum _{i=0}^{n-1}e_{i}x^{i}} 8025:), it follows that it "inherits" all its roots. 7893:{\displaystyle s(x)=\sum _{i=0}^{n-1}c_{i}x^{i}} 7813:, is viewed as the coefficients of a polynomial 6694: 6120: 5839:{\displaystyle s(x)=p(x)\cdot x^{t}-s_{r}(x)\,.} 4946: 759:missions, where they perform within about 1–1.5 340:bit-error correcting codes, since a sequence of 22544:Octave implementation in communications package 21713: 21711: 17669:% = rsDecoder(enc_msg, 8, 301, 12, numel(msg)) 10151:{\displaystyle (1-X_{k}^{-1}\cdot X_{k})=1-1=0} 8382:and nonzero if there is an error. If there are 3119:, and in the classical encoding procedure, its 21871: 21797: 21559: 12595:Find the roots of the error locator polynomial 10080:then one of the multiplied terms will be zero 8578:{\displaystyle \alpha ^{1}\dots \alpha ^{n-k}} 7248:, then by definition all non-zero elements of 574:Reed–Solomon error correction is also used in 541:Reed–Solomon coding is a key component of the 328:erroneous symbols at unknown locations. As an 22459:Algebraic soft-decoding of Reed–Solomon codes 22075:Franke, Steven J.; Taylor, Joseph H. (2016). 21634:New Algorithm For Decoding Reed-Solomon Codes 19274: 19261: 18860:% Put the error magnitude on the error vector 18152:% order to find the error locating polynomial 12681: 12678:is an efficient implementation of this step. 3115:. In other words, the Reed–Solomon code is a 1520: 1444: 778:Channel coding schemes used by NASA missions 594:Almost all two-dimensional bar codes such as 370:Reed–Solomon codes were developed in 1960 by 207: 22441: 22142: 22074: 22035: 21807:Notices of the American Mathematical Society 21708: 21179:702 x + 845 x + 691 x + 461 x + 327 x + 237 21033: 20998: 17456:% get space to add the redundant information 8497:), and the error values at those positions ( 7332: 7308: 4105: 4055: 3310: 3286: 693:Space Communications Protocol Specifications 641:. The encoding process assumes a code of RS( 22348: 22246: 22193:"Shift-register synthesis and BCH decoding" 22170:Lin, Shu; Costello, Jr., Daniel J. (1983), 22169: 21760: 21560:Peterson, W. Wesley; Weldon, E.J. (1996) . 21490: 19189: 14253: 13477:The syndromes are calculated by evaluating 12197:was chosen to be any integer between 1 and 9809: 7910:) is divisible by the generator polynomial 4162:. The sender computes a related polynomial 2075: 1985: 1197: 917: 715:("constraint length" = 7, code rate = 1/2). 383: 22524:Schifra Open Source C++ Reed–Solomon Codec 19366: 17537:% same as the rsgenpoly function on matlab 17162:Decoding beyond the error-correction bound 13742: 13700: 6711:{\textstyle t={\frac {1}{2}}(d_{\min }-1)} 4816: 4808: 3771:Discrete Fourier transform and its inverse 1556: 1533: 1525: 1441: 281:systems such as satellite communications, 214: 200: 22464:Wikiversity:Reed–Solomon codes for coders 22352:(October 1965), "On Decoding BCH Codes", 22307: 22287: 22051: 22015: 21970: 21923: 21887: 21684:Reed Solomon Codes and Their Applications 21656:Reed–Solomon Codes and Their Applications 21612: 17486:% Reed-Solomon generating polynomial g(x) 16261:(0) is the constant (low order) term of A 14103:To calculate the error values, apply the 13185: 13065: 9727:), this is the end. The error locations ( 9708:Consequently, the problem is finding the 6817:{\displaystyle h={\frac {m}{\log _{2}M}}} 6065: 6052: 6051: 5832: 5205:are a subsequence of the coefficients of 2258: 1564: 582:. The distributed online storage service 22412: 22223: 21956: 21536: 19586:{\displaystyle b_{i}E(a_{i})-Q(a_{i})=0} 18998:% Remove leading zeros from Galois array 16500:) as the discrete Fourier transforms of 16440:Decoder using discrete Fourier transform 8515:) can be calculated and subtracted from 7194:see 'Remarks' at the end of this section 6314: 6312:is the number of erasures in the block. 5354:check symbols, dividing that product by 1710:points but are not equal, and thus, the 706: 443: 22373: 22355:IEEE Transactions on Information Theory 22277:The Original View of Reed–Solomon Codes 22201:IEEE Transactions on Information Theory 22039:IEEE Transactions on Information Theory 21875:IEEE Transactions on Information Theory 18896:% Bring this vector to the galois field 14077:Λ(x) = 329 x + 821 x + 001, with roots 7638:{\displaystyle \alpha ^{q}=\alpha ^{1}} 2411:. Thus the classical encoding function 726:Voyager introduced Reed–Solomon coding 515:Guruswami–Sudan list decoding algorithm 14: 22556: 22473: 22419:Error-Control Coding for Data Networks 22229:IRE Transactions on Information Theory 22187: 21650: 21407:= {001, 006, 017, 034, 057, 086, 121} 20864:= {001, 006, 017, 034, 057, 086, 121} 19513:= {001, 006, 123, 456, 057, 086, 121} 19494:= {001, 006, 017, 034, 057, 086, 121} 19184: 17381:% Example: msg = uint8('Test') 11559:that are unaffected by the summation. 10201:{\displaystyle \Lambda (X_{k}^{-1})=0} 7649:of the original codeword derived from 5118:are chosen exactly in such a way that 5089:, and the lower-order coefficients of 477:Bose–Chaudhuri–Hocquenghem (BCH) codes 22328: 22273: 22124:, Stanford University, archived from 21626: 21624: 21400:resulting in the corrected codeword: 21277:) by most significant coefficient of 20857:resulting in the corrected codeword: 12791:) to get the originally sent message 8523:) to get the originally sent message 8224:) to produce the received polynomial 7715:Peterson–Gorenstein–Zierler algorithm 7338:{\displaystyle i\in \{1,\dots ,q-1\}} 6167:maximum distance separable (MDS) code 702: 22498: 22396:The Theory of Error-Correcting Codes 22145:"Simple Algorithms for BCH Decoding" 22114: 21190:266 x + 086 x + 798 x + 311 x + 532 17369:% m is the number of bits per symbol 14590: 14110:Ω(x) = S(x) Λ(x) mod x = 546 x + 732 12588: 11215:Collect each term into its own sum. 8971:, as shown in the previous section. 8534: 7700: 7528:{\displaystyle a\mapsto p(\alpha a)} 7124:-symbol block; this is denoted as a 3733:{\displaystyle m_{0},\dots ,m_{k-1}} 3422:{\displaystyle a_{k},\dots ,a_{n-1}} 2384:{\displaystyle a_{0},\dots ,a_{n-1}} 1714:of the Reed–Solomon code is exactly 683:Reed–Solomon codes are also used in 624: 22334:"The Ubiquitous Reed–Solomon Codes" 22152:IEEE Transactions on Communications 21630: 19176:Reed Solomon original view decoders 17378:% n is the total size (k+redundant) 14583:is the error locator polynomial, Λ( 14266:) with an assumed number of errors 14246:) reproduces the original codeword 10303:{\displaystyle Y_{k}X_{k}^{j+\nu }} 7709:Peterson–Gorenstein–Zierler decoder 1939:{\displaystyle \delta +R\leq 1+1/n} 457:concatenated error correction codes 432:Berlekamp–Massey decoding algorithm 24: 22512: 22474:Geisel, William A. (August 1990), 22108: 21621: 21039:{\displaystyle \{a_{i},b(a_{i})\}} 19265: 17052: 17017: 16988: 16932: 16897: 16869: 15833: 15816: 15792: 15641: 15592: 15569: 15520: 15497: 15474: 15425: 15402: 15379: 15330: 15307: 15284: 15235: 15212: 15189: 15153: 15130: 15094: 15046: 14994: 14962: 14946: 14905: 14882: 14862: 14840: 14799: 14776: 14756: 14335: 14297: 14277: 14019: 14005: 13856: 13842: 12452: 12425: 12411: 12211:of the error location polynomial: 12150: 12103: 12074: 11986: 11951: 11910: 11800: 11713: 11632: 11546: 11482: 11395: 11314: 11158: 11102: 11052: 10920: 10864: 10814: 10710: 10642: 10580: 10502: 10465: 10434: 10353: 10165: 9965: 9936: 9913: 9842: 6588: 6511:and for other modulation schemes: 6436: 5040:such that the coefficients of the 3182:. In this variant, the polynomial 2047:{\displaystyle a_{1},\dots ,a_{k}} 1335:{\displaystyle a_{1},\dots ,a_{n}} 1284:elements. In turn, the polynomial 430:, and has since been known as the 390:(unencoded message length) out of 25: 22580: 22436: 19595:Assume maximum number of errors: 17330: 17170:states that the minimum distance 16611:) are the same as the syndromes: 12753: 12704:matrix given above, or using the 8017:) is a multiple of the generator 7199:Whether the Reed–Solomon code is 6769:{\displaystyle P_{s}=1-(1-s)^{h}} 5533:The remainder has degree at most 4004:{\displaystyle a_{i}=\alpha ^{i}} 885:Cassini, Mars Pathfinder, others 378:, who were then staff members of 21957:Guo, Zeyu; Zhang, Zihan (2023). 21423:Berlekamp–Massey algorithm 19371:In 1986, a decoder known as the 17804:% polynomial the result is zero) 17296: 17174:of a linear block code of size ( 16476:) are the same as above. Define 14383:Berlekamp–Massey algorithm 14260:Berlekamp–Massey algorithm 10255:{\displaystyle 1\leq j\leq \nu } 8908:{\displaystyle s(\alpha ^{j})=0} 8490:), the positions of the errors ( 7184:The Reed–Solomon code, like the 6249:. A Reed–Solomon code (like any 5885: 5714:has the property that the first 4744: 4122: 2450:{\displaystyle C:F^{k}\to F^{n}} 2074:In the original construction of 1948:maximum distance separable codes 1752:. Then the relative distance is 1582:polynomials of degree less than 1539: is a polynomial over 1427: 1405: 1123:{\displaystyle R={\frac {k}{n}}} 22093:from the original on 2017-03-09 22068: 22029: 21997: 21950: 21903: 21865: 21856: 21844: 21835: 21820:from the original on 2008-05-09 21791: 21780:from the original on 2019-02-01 21766: 21753: 21460: 19599:= 2. The key equations become: 17344:implementation for an encoder. 16778:) using the known coefficients 16676: 11539:Extract the constant values of 9293:The syndromes give a system of 8979:Error locators and error values 8825: 7167:{\displaystyle (n,k)=(255,223)} 6165:. Such a code is also called a 6047: 5678:{\displaystyle p(x)\cdot x^{t}} 2004:of the polynomial at the first 898:Messenger, Stereo, MRO, others 800:Explorer, Mariner, many others 533: 528: 190:Maximum-distance separable code 22564:Error detection and correction 22469:BBC R&D White Paper WHP031 21675: 21644: 21588: 21553: 21530: 21495: 21484: 21030: 21017: 20947: 20928: 20868: 19780: 19660: 19654: 19618: 19574: 19561: 19552: 19539: 19399: 19386: 19350: 19338: 19306: 19294: 17375:% k is the size of the message 17263: 17250: 17077: 16984: 16957: 16880: 16673: 16660: 15929:increases. Once the degree of 15055: 15049: 15030: 15024: 15015: 15009: 15003: 14997: 14871: 14865: 14765: 14759: 14646: 14640: 14254:Berlekamp–Massey decoder 13772: 13759: 13730: 13717: 13688: 13675: 13521: 13508: 13369: 13363: 13354: 13348: 13339: 13333: 13215: 13209: 13182: 13176: 13167: 13161: 13092: 13086: 13062: 13056: 13047: 13041: 12932: 12913: 12910: 12891: 12888: 12869: 12866: 12854: 12848: 12842: 10377: 10356: 10189: 10168: 10127: 10087: 9900: 9878: 9851: 9845: 9253: 9232: 9187: 9173: 8975:was not corrupted in transit. 8931: 8925: 8896: 8883: 8739: 8726: 8710: 8697: 8682: 8669: 8660: 8647: 8638: 8625: 8413: 8407: 8306: 8300: 8284:{\displaystyle r(x)=s(x)+e(x)} 8278: 8272: 8263: 8257: 8248: 8242: 8156: 8143: 8114: 8093: 8085: 8079: 8070: 8049: 8041: 8035: 7986: 7967: 7934: 7928: 7837: 7831: 7800: 7743: 7731: 7592: 7589: 7576: 7561: 7548: 7542: 7522: 7513: 7507: 7484: 7481: 7462: 7447: 7434: 7428: 7367: 7359: 7161: 7149: 7143: 7131: 7022:computer systems.) The number 6757: 6744: 6705: 6686: 6638: 6618: 6486: 6466: 6217: 6205: 6062: 6056: 6038: 6032: 6016: 6010: 5994: 5988: 5959: 5953: 5944: 5938: 5915: 5909: 5864: 5858: 5829: 5823: 5794: 5788: 5779: 5773: 5750: 5744: 5701: 5695: 5659: 5653: 5559:, whereas the coefficients of 5515: 5509: 5477: 5471: 5462: 5456: 5426: 5420: 5370: 5364: 5282: 5276: 5250: 5244: 5221: 5215: 5192: 5186: 5163: 5157: 5134: 5128: 5105: 5099: 5076: 5070: 5027: 5021: 4998: 4992: 4986: 4980: 4971: 4965: 4826: 4820: 4484: 4478: 4436: 4430: 4381: 4375: 4323: 4317: 4294: 4288: 4265: 4259: 4178: 4172: 4149: 4143: 3932: 3913: 3889: 3876: 3859: 3846: 3822: 3816: 3675: 3656: 3634: 3621: 3557: 3538: 3514: 3501: 3484: 3471: 3447: 3441: 3259: 3246: 2748: 2742: 2644: 2638: 2583: 2564: 2540: 2527: 2510: 2497: 2473: 2467: 2434: 2205: 2199: 2129: 2091: 1653: 1641: 1515: 1502: 1487: 1474: 1465: 1452: 852:Voyager, Galileo, many others 494:, but BCH codes are used with 289:, and storage systems such as 13: 1: 21631:Gao, Shuhong (January 2002), 21614:10.1016/S0019-9958(75)90090-X 21477: 19464:If the message polynomial is 17723:% Initialize the error vector 16770:= number of errors. Generate 14238:from the received polynomial 13005:If the message polynomial is 12712:Calculate the error locations 7011:{\displaystyle n=2^{8}-1=255} 6146:{\textstyle d_{\min }=n-k+1.} 6081: 5004:{\displaystyle s(x)=p(x)g(x)} 4953:systematic encoding procedure 4947:Systematic encoding procedure 1677:{\displaystyle n-(k-1)=n-k+1} 22499:Reid, Jeff A. (April 1995), 21981:10.1109/FOCS57990.2023.00019 21763:, p. 171), for example. 21537:Peterson, W. Wesley (1961). 20876: 19435: 18293:% Update g as g0-quotient*g1 16268: 14597:extended Euclidean algorithm 14400: 12783:and then is subtracted from 10073:{\displaystyle x=X_{k}^{-1}} 8983:For convenience, define the 7645:holds, this codeword is the 6292:is the number of errors and 5892:{\displaystyle \mathbf {C} } 2155:is mapped to the polynomial 1412:{\displaystyle \mathbf {C} } 1020:{\displaystyle k<n\leq q} 523:extended Euclidean algorithm 439:extended Euclidean algorithm 7: 22118:EE387 Notes #7, Handout #28 21686:. IEEE Press. p. 433. 21566:(2nd ed.). MIT Press. 21411: 17658:encoded, m, prim_poly, n, k 16179:to set low order term of Λ( 10984:and it will still be zero. 10310:and it will still be zero. 8964:{\displaystyle \alpha ^{j}} 7288:{\displaystyle \alpha ^{i}} 5848:As a result, the codewords 5176:. Then the coefficients of 4708:For a "narrow sense code", 3326:To compute this polynomial 763:of the ultimate limit, the 589: 10: 22585: 21853:by W Wesley Peterson, 1961 20991:Lagrange interpolation of 18218:% Find quotient*g1 and pad 17640: 17335: 12802: 12799:), with errors corrected. 12682:Calculate the error values 12667:{\displaystyle X_{k}^{-1}} 12049:{\displaystyle S_{j+\nu }} 10032:{\displaystyle X_{k}^{-1}} 8386:errors at distinct powers 7712: 7680: 3777:discrete Fourier transform 2292:is obtained by evaluating 2054:and obtain the polynomial 871:RS(255, 223) + CC(14, 1/4) 481:Digital Video Broadcasting 365: 22442:Information and tutorials 21739:10.1109/JPROC.2007.905132 21491:Reed & Solomon (1960) 21428:Berlekamp–Welch algorithm 19373:Berlekamp–Welch algorithm 19356:{\displaystyle (255,249)} 19190:Reed & Solomon (1960) 18791:% Solve the linear system 17340:Here we present a simple 17269:{\displaystyle GF(2^{m})} 17231:algebraic geometric codes 14393:) is used to represent Λ( 10262:. Multiply both sides by 10230:be any integer such that 9760:errors can be corrected. 7736:The transmitted message, 7421:gives rise to a codeword 6966:{\displaystyle n=2^{m}-1} 6864:is the modulation order. 4249:and sends the polynomial 4242:{\displaystyle n\leq q-1} 2685:{\displaystyle n\times k} 2076:Reed & Solomon (1960) 1986:Reed & Solomon (1960) 1198:Reed & Solomon (1960) 860:RS(255, 223) + CC(7, 1/3) 849:RS(255, 223) + CC(7, 1/2) 507:Berlekamp–Welch algorithm 320:locate and correct up to 195: 188: 183: 168: 163: 148: 113: 96: 84: 72: 56: 51: 37: 32: 22502:CRC and Reed Solomon ECC 22480:, Technical Memorandum, 22416:; Chen, Xuemin (2012) , 22368:10.1109/TIT.1965.1053825 22241:10.1109/TIT.1960.1057586 22210:10.1109/tit.1969.1054260 21761:Lin & Costello (1983 21453: 21448:Forward error correction 21443:Folded Reed–Solomon code 21084:{\displaystyle A_{-1}=0} 19460:= {0, 1, 2, 3, 4, 5, 6} 19405:{\displaystyle O(n^{3})} 17647: 17346: 17314:, which employ iterated 16460:) the encoded codeword. 12686:Once the error locators 11552:{\displaystyle \Lambda } 9826:error locator polynomial 9810:Error locator polynomial 8006:is a primitive element. 6161:); this is known as the 5432:{\displaystyle s_{r}(x)} 2628:, that is, it satisfies 1244:, over the finite field 918:Constructions (encoding) 697:forward error correction 487:, in conjunction with a 233:that were introduced by 22313:Algebraic Coding Theory 22062:10.1109/TIT.2003.819332 21934:10.1145/3564246.3585128 21727:Proceedings of the IEEE 21601:Information and Control 21118:{\displaystyle A_{0}=1} 20849:E(x) = 0 : {2, 3} 19517:The key equations are: 19367:Berlekamp Welch decoder 17645:Now the decoding part: 17361:msg, m, prim_poly, n, k 12736:{\displaystyle \alpha } 12723:by taking the log base 12639:(not their reciprocals 12056:from both sides yields 8594:. They are defined as: 7217:{\displaystyle \alpha } 6231:{\displaystyle (n-k)/2} 5877:are indeed elements of 4462:{\displaystyle \alpha } 2659:{\displaystyle C(m)=Am} 1988:interprets the message 1745:{\displaystyle d=n-k+1} 384:Reed & Solomon 1960 332:, it can correct up to 22374:Koetter, Ralf (2005), 22293:Nonbinary BCH decoding 22204:, IT-15 (1): 122–127, 21851:Error Correcting Codes 21563:Error Correcting Codes 21539:Error Correcting Codes 21119: 21085: 21040: 20985: 20984:{\displaystyle R_{0}=} 20954: 20927: 20751: 20281: 19818: 19587: 19426: 19406: 19357: 19325: 19240: 19213: 18434:% Remove leading zeros 17270: 17182:) is upper-bounded by 17110: 16826: 16799: 16757: 16730: 16701: 15854: 15062: 14979: 14367: 14232: 14069: 13941: 13785: 13647: 13471: 13317: 13146: 12999: 12737: 12668: 12633: 12599:Use the coefficients Λ 12564: 12185: 12050: 12015: 11875: 11834: 11747: 11666: 11591: 11553: 11531: 11480: 11393: 11312: 11247: 11207: 11011: 10968: 10304: 10256: 10224: 10202: 10152: 10074: 10033: 9988: 9877: 9800: 9786:, instead of just the 9780: 9754: 9700: 9284: 9158: 9128: 9080: 8965: 8938: 8909: 8867: 8772: 8579: 8478: 8439: 8363: 8338: 8285: 8208: 8127: 7996: 7966: 7894: 7869: 7807: 7722:Error Correcting Codes 7663: 7639: 7599: 7529: 7491: 7415: 7395: 7375: 7339: 7289: 7262: 7242: 7218: 7168: 7118: 7092: 7062: 7061:{\displaystyle k<n} 7036: 7012: 6967: 6928: 6908: 6881: 6858: 6838: 6818: 6770: 6712: 6657: 6577: 6505: 6425: 6320: 6306: 6286: 6243:signal-to-noise ratios 6232: 6192: 6147: 6073: 5922: 5893: 5871: 5840: 5757: 5728: 5708: 5679: 5637: 5553: 5525: 5433: 5397: 5377: 5348: 5316: 5289: 5257: 5228: 5199: 5170: 5141: 5112: 5083: 5054: 5034: 5005: 4938: 4852: 4728: 4700: 4463: 4443: 4414: 4388: 4356: 4330: 4301: 4272: 4243: 4211: 4185: 4156: 4112: 4005: 3947: 3800: 3761: 3734: 3682: 3598: 3572: 3423: 3375:Lagrange interpolation 3367: 3347: 3320: 3223: 3203: 3176: 3137: 3109: 3083: 2726: 2706: 2686: 2660: 2618: 2598: 2451: 2405: 2385: 2333: 2313: 2286: 2266: 2237: 2176: 2149: 2048: 1978: 1940: 1886: 1852: 1746: 1704: 1678: 1622: 1596: 1572: 1413: 1336: 1278: 1258: 1234: 1214: 1182: 1150: 1124: 1084: 1064: 1044: 1021: 986: 963: 940: 753:Mars Exploration Rover 716: 448: 380:MIT Lincoln Laboratory 231:error-correcting codes 27:Error-correcting codes 22274:Welch, L. R. (1997), 21120: 21086: 21041: 20986: 20955: 20907: 20752: 20282: 19819: 19588: 19427: 19407: 19358: 19326: 19241: 19239:{\displaystyle a_{n}} 19214: 19212:{\displaystyle a_{1}} 17271: 17111: 16827: 16825:{\displaystyle E_{t}} 16800: 16798:{\displaystyle E_{1}} 16758: 16756:{\displaystyle R_{t}} 16731: 16729:{\displaystyle R_{1}} 16702: 15855: 15063: 14987:The key equation is: 14980: 14368: 14233: 14070: 13942: 13786: 13648: 13472: 13318: 13147: 13000: 12738: 12669: 12634: 12632:{\displaystyle X_{k}} 12565: 12186: 12051: 12016: 11876: 11814: 11727: 11646: 11571: 11554: 11532: 11460: 11373: 11292: 11227: 11208: 10991: 10969: 10305: 10257: 10225: 10203: 10153: 10075: 10034: 9989: 9857: 9801: 9781: 9755: 9701: 9285: 9159: 9108: 9081: 8966: 8939: 8910: 8868: 8752: 8580: 8479: 8419: 8364: 8312: 8286: 8209: 8128: 7997: 7940: 7895: 7843: 7808: 7664: 7640: 7600: 7530: 7497:. Since the function 7492: 7416: 7396: 7376: 7374:{\displaystyle q=|F|} 7340: 7290: 7263: 7243: 7219: 7169: 7119: 7117:{\displaystyle n=255} 7093: 7091:{\displaystyle k=223} 7063: 7037: 7013: 6968: 6929: 6909: 6907:{\displaystyle 2^{m}} 6882: 6859: 6839: 6819: 6771: 6713: 6658: 6551: 6506: 6399: 6318: 6307: 6287: 6233: 6193: 6148: 6074: 5923: 5894: 5872: 5841: 5758: 5729: 5709: 5680: 5638: 5554: 5526: 5434: 5398: 5378: 5349: 5347:{\displaystyle t=n-k} 5322:to make room for the 5317: 5315:{\displaystyle x^{t}} 5290: 5258: 5229: 5200: 5171: 5147:becomes divisible by 5142: 5113: 5084: 5055: 5035: 5006: 4939: 4832: 4729: 4701: 4464: 4444: 4415: 4389: 4357: 4331: 4302: 4273: 4244: 4212: 4186: 4157: 4113: 4006: 3948: 3801: 3799:{\displaystyle p_{m}} 3762: 3760:{\displaystyle p_{m}} 3740:by the definition of 3735: 3683: 3599: 3573: 3424: 3368: 3348: 3346:{\displaystyle p_{m}} 3321: 3224: 3204: 3202:{\displaystyle p_{m}} 3177: 3175:{\displaystyle p_{m}} 3138: 3110: 3084: 2727: 2707: 2687: 2661: 2619: 2599: 2452: 2406: 2386: 2334: 2314: 2312:{\displaystyle p_{m}} 2287: 2267: 2211: 2177: 2175:{\displaystyle p_{m}} 2150: 2049: 1979: 1977:{\displaystyle q^{k}} 1941: 1887: 1885:{\displaystyle R=k/n} 1853: 1747: 1705: 1679: 1623: 1597: 1573: 1547: of degree 1414: 1337: 1279: 1259: 1235: 1215: 1183: 1181:{\displaystyle n=q-1} 1151: 1125: 1085: 1065: 1045: 1022: 987: 964: 941: 710: 657:symbols each storing 473:digital communication 447: 269:technologies such as 22389:MacWilliams, F. J.; 21392:) = 0 : {2, 3} 21285:) = 708. (Optional) 21193:708 x + 176 x + 532 21096: 21059: 20995: 20965: 20888: 20299: 20291:Gaussian elimination 19829: 19605: 19523: 19416: 19380: 19335: 19255: 19249:binomial coefficient 19223: 19196: 18161:% Do a long division 17276:and its extensions. 17241: 17226:Venkatesan Guruswami 16836: 16809: 16782: 16740: 16713: 16615: 15921:where the degree of 15082: 14991: 14630: 14274: 14150: 13957: 13951:Gaussian elimination 13794: 13656: 13489: 13327: 13155: 13028: 12836: 12766:) is generated from 12727: 12643: 12616: 12215: 12060: 12027: 11887: 11563: 11543: 11219: 10988: 10314: 10266: 10234: 10214: 10162: 10084: 10043: 10008: 10004:are the reciprocals 9839: 9790: 9767: 9738: 9336: 9170: 9092: 9007: 8948: 8937:{\displaystyle s(x)} 8919: 8877: 8598: 8543: 8401: 8294: 8236: 8137: 8029: 7922: 7825: 7740: 7675:BCH codes are cyclic 7653: 7609: 7539: 7501: 7425: 7405: 7385: 7349: 7299: 7272: 7252: 7232: 7208: 7128: 7102: 7076: 7046: 7026: 6977: 6938: 6918: 6891: 6871: 6848: 6828: 6780: 6722: 6667: 6515: 6335: 6296: 6276: 6272:is satisfied, where 6202: 6176: 6112: 5932: 5921:{\displaystyle g(x)} 5903: 5881: 5870:{\displaystyle s(x)} 5852: 5767: 5756:{\displaystyle p(x)} 5738: 5718: 5707:{\displaystyle s(x)} 5689: 5647: 5563: 5537: 5443: 5407: 5387: 5376:{\displaystyle g(x)} 5358: 5326: 5299: 5288:{\displaystyle p(x)} 5270: 5256:{\displaystyle p(x)} 5238: 5227:{\displaystyle s(x)} 5209: 5198:{\displaystyle p(x)} 5180: 5169:{\displaystyle g(x)} 5151: 5140:{\displaystyle s(x)} 5122: 5111:{\displaystyle s(x)} 5093: 5082:{\displaystyle p(x)} 5064: 5044: 5033:{\displaystyle s(x)} 5015: 4959: 4740: 4712: 4472: 4453: 4442:{\displaystyle g(x)} 4424: 4398: 4387:{\displaystyle g(x)} 4369: 4364:generator polynomial 4340: 4329:{\displaystyle p(x)} 4311: 4300:{\displaystyle s(x)} 4282: 4271:{\displaystyle s(x)} 4253: 4221: 4195: 4184:{\displaystyle s(x)} 4166: 4155:{\displaystyle p(x)} 4137: 4014: 3975: 3810: 3783: 3744: 3692: 3608: 3588: 3435: 3381: 3357: 3330: 3233: 3213: 3186: 3159: 3127: 3099: 2736: 2716: 2696: 2670: 2632: 2608: 2461: 2415: 2395: 2343: 2323: 2296: 2276: 2186: 2159: 2082: 2012: 1961: 1904: 1862: 1756: 1718: 1688: 1632: 1606: 1586: 1423: 1401: 1300: 1268: 1248: 1240:of degree less than 1224: 1204: 1160: 1134: 1101: 1074: 1054: 1034: 999: 976: 953: 930: 653:codewords of length 22495:(scholarpedia.org). 22330:Cipra, Barry Arthur 22309:Berlekamp, Elwyn R. 22289:Berlekamp, Elwyn R. 22225:Peterson, Wesley W. 22115:Gill, John (n.d.), 19813: 19779: 19751: 19723: 19185:Theoretical decoder 19108:% Add leading zeros 14579:The final value of 14373:and then adjusts Λ( 14113:Λ'(x) = 658 x + 821 12663: 11859: 11784: 11703: 11622: 11515: 11440: 11359: 11278: 11191: 11147: 11097: 11047: 10953: 10909: 10859: 10809: 10767: 10749: 10699: 10681: 10637: 10619: 10575: 10528: 10491: 10460: 10421: 10376: 10352: 10299: 10188: 10113: 10069: 10028: 9753:{\displaystyle n-k} 9556: 9528: 9505: 9458: 9436: 9419: 9400: 9378: 9361: 9279: 9153: 6617: 6465: 6245:)—these are called 6191:{\displaystyle n-k} 5552:{\displaystyle t-1} 4727:{\displaystyle i=1} 4413:{\displaystyle n-k} 4355:{\displaystyle k-1} 4336:, which has degree 4210:{\displaystyle n-1} 3277: for all 3003: 2969: 2894: 2866: 2830: 2802: 2712:with elements from 1703:{\displaystyle k-1} 1621:{\displaystyle k-1} 1149:{\displaystyle n=q} 911:Constellation, MSL 809:convolutional codes 779: 732:convolutional codes 664:Any combination of 649:) which results in 316:erroneous symbols, 22489:Concatenated codes 22377:Reed–Solomon Codes 22086:(May/June): 8–17. 21115: 21081: 21036: 20981: 20950: 20747: 20741: 20677: 20567: 20277: 20271: 20207: 20097: 19814: 19799: 19765: 19737: 19709: 19583: 19422: 19402: 19353: 19321: 19236: 19209: 17266: 17106: 17104: 16822: 16795: 16753: 16726: 16697: 15850: 15844: 15681: 15058: 14975: 14973: 14363: 14228: 14065: 14059: 14030: 13990: 13937: 13931: 13902: 13867: 13827: 13781: 13643: 13467: 13313: 13142: 12995: 12733: 12664: 12646: 12629: 12560: 12554: 12463: 12396: 12181: 12046: 12011: 11871: 11845: 11758: 11677: 11602: 11549: 11527: 11501: 11414: 11333: 11258: 11203: 11177: 11121: 11071: 11027: 10964: 10962: 10939: 10883: 10833: 10789: 10750: 10729: 10682: 10661: 10620: 10599: 10555: 10511: 10474: 10443: 10401: 10359: 10332: 10300: 10279: 10252: 10220: 10198: 10171: 10148: 10096: 10070: 10052: 10029: 10011: 9984: 9796: 9779:{\displaystyle 2v} 9776: 9750: 9696: 9690: 9620: 9559: 9536: 9508: 9485: 9444: 9422: 9405: 9386: 9364: 9347: 9280: 9265: 9154: 9139: 9076: 8961: 8934: 8905: 8863: 8861: 8575: 8474: 8359: 8281: 8204: 8123: 7992: 7890: 7803: 7726:W. Wesley Peterson 7659: 7635: 7595: 7525: 7487: 7411: 7391: 7381:. Each polynomial 7371: 7335: 7285: 7258: 7238: 7214: 7186:convolutional code 7164: 7114: 7088: 7058: 7032: 7008: 6963: 6924: 6904: 6877: 6854: 6834: 6814: 6766: 6708: 6653: 6603: 6501: 6451: 6321: 6302: 6282: 6228: 6188: 6143: 6069: 5918: 5889: 5867: 5836: 5753: 5724: 5704: 5675: 5643:in the polynomial 5633: 5549: 5521: 5429: 5393: 5373: 5344: 5312: 5285: 5253: 5224: 5195: 5166: 5137: 5108: 5079: 5050: 5030: 5001: 4934: 4724: 4696: 4459: 4439: 4410: 4384: 4352: 4326: 4297: 4268: 4239: 4207: 4181: 4152: 4108: 4001: 3943: 3937: 3796: 3757: 3730: 3678: 3594: 3568: 3562: 3419: 3363: 3343: 3316: 3219: 3199: 3172: 3133: 3105: 3093:Vandermonde matrix 3079: 3073: 3006: 2977: 2949: 2874: 2852: 2810: 2788: 2722: 2702: 2682: 2666:for the following 2656: 2614: 2594: 2588: 2447: 2401: 2381: 2329: 2309: 2282: 2262: 2172: 2145: 2044: 1996:of the polynomial 1974: 1936: 1882: 1848: 1742: 1700: 1674: 1618: 1592: 1568: 1409: 1397:Formally, the set 1332: 1274: 1254: 1230: 1210: 1178: 1146: 1120: 1080: 1060: 1040: 1017: 982: 959: 936: 777: 717: 703:Space transmission 498:in its successor, 449: 413:W. Wesley Peterson 227:Reed–Solomon codes 33:Reed–Solomon codes 22406:978-0-444-85010-2 22399:, North-Holland, 22322:978-0-89412-063-3 22181:978-0-13-283796-5 22174:, Prentice-Hall, 22164:10.1109/26.403765 22046:(11): 2809–2825. 21990:979-8-3503-1894-4 21943:978-1-4503-9913-5 21898:10.1109/18.782097 21669:978-0-7803-1025-4 21573:978-0-585-30709-1 21197: 21196: 19425:{\displaystyle n} 19319: 19272: 17320:theoretical limit 17085: 16878: 16680: 16397: 16396: 16183:) to 1, divide Λ( 14591:Euclidean decoder 14577: 14576: 14346: 14308: 12828:(this is used in 10223:{\displaystyle j} 9799:{\displaystyle v} 9045: 8535:Syndrome decoding 8170: 7701:BCH view decoders 7662:{\displaystyle p} 7647:cyclic left-shift 7414:{\displaystyle F} 7394:{\displaystyle p} 7261:{\displaystyle F} 7241:{\displaystyle F} 7035:{\displaystyle k} 6927:{\displaystyle m} 6880:{\displaystyle F} 6857:{\displaystyle M} 6837:{\displaystyle s} 6812: 6684: 6595: 6549: 6539: 6443: 6397: 6387: 6305:{\displaystyle S} 6285:{\displaystyle E} 6088:linear block code 5727:{\displaystyle k} 5503: 5495: 5396:{\displaystyle t} 5053:{\displaystyle k} 4876: 4278:. The polynomial 3597:{\displaystyle k} 3366:{\displaystyle m} 3278: 3222:{\displaystyle k} 3136:{\displaystyle A} 3108:{\displaystyle F} 3091:This matrix is a 2725:{\displaystyle F} 2705:{\displaystyle A} 2617:{\displaystyle C} 2404:{\displaystyle F} 2339:different points 2332:{\displaystyle n} 2285:{\displaystyle m} 1602:agree in at most 1595:{\displaystyle k} 1548: 1540: 1388:primitive element 1277:{\displaystyle q} 1257:{\displaystyle F} 1233:{\displaystyle p} 1213:{\displaystyle k} 1118: 1083:{\displaystyle q} 1063:{\displaystyle q} 1043:{\displaystyle F} 985:{\displaystyle k} 962:{\displaystyle n} 939:{\displaystyle q} 915: 914: 882:RS + CC (15, 1/6) 837:Binary Golay code 625:Data transmission 408:Daniel Gorenstein 267:data transmission 224: 223: 68:Reed–Solomon code 61:Linear block code 16:(Redirected from 22576: 22508: 22507: 22485: 22432: 22409: 22391:Sloane, N. J. A. 22385: 22370: 22345: 22325: 22304: 22284: 22282: 22270: 22252:Solomon, Gustave 22243: 22220: 22197: 22184: 22166: 22149: 22139: 22138: 22136: 22131:on June 30, 2014 22130: 22123: 22102: 22101: 22099: 22098: 22092: 22081: 22072: 22066: 22065: 22055: 22033: 22027: 22026: 22025: 22024: 22019: 22001: 21995: 21994: 21974: 21954: 21948: 21947: 21927: 21907: 21901: 21900: 21891: 21882:(6): 1757–1767, 21869: 21863: 21860: 21854: 21848: 21842: 21839: 21833: 21827: 21826: 21825: 21819: 21804: 21795: 21789: 21788: 21786: 21785: 21770: 21764: 21757: 21751: 21750: 21724: 21715: 21706: 21705: 21679: 21673: 21672: 21652:Immink, K. A. S. 21648: 21642: 21641: 21639: 21628: 21619: 21618: 21616: 21592: 21586: 21585: 21557: 21551: 21550: 21534: 21528: 21527: 21499: 21493: 21488: 21471: 21464: 21408: 21399: 21393: 21382: 21369: 21333: 21313: 21258: 21231: 21128: 21127: 21124: 21122: 21121: 21116: 21108: 21107: 21090: 21088: 21087: 21082: 21074: 21073: 21045: 21043: 21042: 21037: 21029: 21028: 21010: 21009: 20990: 20988: 20987: 20982: 20977: 20976: 20959: 20957: 20956: 20951: 20946: 20945: 20926: 20921: 20903: 20902: 20865: 20856: 20850: 20846: 20840: 20804: 20784: 20756: 20754: 20753: 20748: 20746: 20745: 20682: 20681: 20674: 20673: 20660: 20659: 20646: 20645: 20632: 20631: 20618: 20617: 20604: 20603: 20590: 20589: 20572: 20571: 20286: 20284: 20283: 20278: 20276: 20275: 20212: 20211: 20204: 20203: 20190: 20189: 20176: 20175: 20162: 20161: 20148: 20147: 20134: 20133: 20120: 20119: 20102: 20101: 19823: 19821: 19820: 19815: 19812: 19807: 19798: 19797: 19778: 19773: 19764: 19763: 19750: 19745: 19736: 19735: 19722: 19717: 19708: 19707: 19695: 19694: 19685: 19684: 19672: 19671: 19653: 19652: 19643: 19642: 19630: 19629: 19617: 19616: 19592: 19590: 19589: 19584: 19573: 19572: 19551: 19550: 19535: 19534: 19514: 19495: 19487:The codeword is 19484: 19461: 19431: 19429: 19428: 19423: 19411: 19409: 19408: 19403: 19398: 19397: 19362: 19360: 19359: 19354: 19330: 19328: 19327: 19322: 19320: 19318: 19292: 19284: 19279: 19278: 19277: 19264: 19245: 19243: 19242: 19237: 19235: 19234: 19218: 19216: 19215: 19210: 19208: 19207: 19171: 19168: 19165: 19162: 19159: 19156: 19153: 19150: 19147: 19144: 19141: 19138: 19135: 19132: 19129: 19126: 19122: 19119: 19116: 19112: 19109: 19106: 19103: 19100: 19097: 19094: 19091: 19088: 19085: 19082: 19079: 19076: 19073: 19070: 19067: 19064: 19061: 19058: 19055: 19052: 19049: 19046: 19043: 19040: 19037: 19034: 19031: 19028: 19025: 19022: 19019: 19016: 19012: 19009: 19006: 19002: 18999: 18996: 18993: 18990: 18987: 18984: 18981: 18978: 18975: 18972: 18969: 18966: 18963: 18960: 18957: 18954: 18951: 18948: 18945: 18942: 18939: 18936: 18933: 18930: 18927: 18924: 18921: 18918: 18915: 18912: 18909: 18906: 18903: 18900: 18897: 18894: 18891: 18888: 18885: 18882: 18879: 18876: 18873: 18870: 18867: 18864: 18861: 18858: 18855: 18852: 18849: 18846: 18843: 18840: 18837: 18834: 18831: 18828: 18825: 18822: 18819: 18816: 18813: 18810: 18807: 18804: 18801: 18798: 18795: 18792: 18789: 18786: 18783: 18780: 18777: 18774: 18771: 18768: 18765: 18762: 18759: 18756: 18753: 18750: 18747: 18744: 18741: 18738: 18735: 18732: 18729: 18726: 18723: 18720: 18717: 18714: 18711: 18708: 18705: 18702: 18699: 18696: 18693: 18690: 18687: 18684: 18681: 18678: 18675: 18672: 18669: 18666: 18663: 18660: 18657: 18654: 18651: 18648: 18645: 18642: 18639: 18636: 18633: 18630: 18627: 18624: 18621: 18618: 18615: 18612: 18609: 18606: 18603: 18600: 18597: 18594: 18591: 18588: 18585: 18582: 18579: 18576: 18573: 18570: 18567: 18564: 18561: 18558: 18555: 18552: 18549: 18546: 18543: 18540: 18537: 18534: 18531: 18528: 18525: 18522: 18519: 18516: 18513: 18510: 18507: 18504: 18501: 18498: 18495: 18492: 18489: 18486: 18483: 18480: 18477: 18474: 18471: 18468: 18465: 18462: 18459: 18456: 18453: 18450: 18447: 18444: 18441: 18438: 18435: 18432: 18429: 18426: 18423: 18420: 18417: 18414: 18411: 18408: 18405: 18402: 18399: 18396: 18393: 18390: 18387: 18384: 18381: 18378: 18375: 18372: 18369: 18366: 18363: 18360: 18357: 18354: 18351: 18348: 18345: 18342: 18339: 18336: 18333: 18330: 18327: 18324: 18321: 18318: 18315: 18312: 18309: 18306: 18303: 18300: 18297: 18294: 18291: 18288: 18285: 18282: 18279: 18276: 18273: 18270: 18267: 18264: 18261: 18258: 18255: 18252: 18249: 18246: 18243: 18240: 18237: 18234: 18231: 18228: 18225: 18222: 18219: 18216: 18213: 18210: 18207: 18204: 18201: 18198: 18195: 18192: 18189: 18186: 18185:% Add some zeros 18183: 18180: 18177: 18174: 18171: 18168: 18165: 18162: 18159: 18156: 18153: 18150: 18147: 18144: 18141: 18138: 18135: 18132: 18129: 18126: 18123: 18120: 18117: 18114: 18111: 18108: 18105: 18102: 18099: 18096: 18093: 18090: 18087: 18084: 18081: 18078: 18075: 18072: 18069: 18066: 18063: 18060: 18057: 18054: 18051: 18048: 18045: 18042: 18039: 18036: 18033: 18030: 18027: 18024: 18021: 18018: 18015: 18012: 18009: 18006: 18003: 18000: 17997: 17994: 17991: 17988: 17985: 17982: 17979: 17976: 17973: 17970: 17967: 17964: 17961: 17958: 17955: 17952: 17949: 17946: 17943: 17940: 17937: 17934: 17931: 17928: 17925: 17922: 17919: 17916: 17913: 17910: 17907: 17904: 17901: 17898: 17895: 17892: 17889: 17886: 17883: 17880: 17877: 17874: 17871: 17868: 17865: 17862: 17859: 17856: 17853: 17850: 17847: 17844: 17841: 17838: 17835: 17832: 17829: 17826: 17823: 17820: 17817: 17814: 17811: 17808: 17805: 17802: 17799: 17796: 17793: 17790: 17787: 17784: 17781: 17778: 17775: 17772: 17769: 17766: 17763: 17760: 17757: 17754: 17751: 17748: 17745: 17742: 17739: 17736: 17733: 17730: 17727: 17724: 17721: 17718: 17715: 17712: 17709: 17706: 17703: 17700: 17697: 17694: 17691: 17688: 17685: 17682: 17679: 17676: 17673: 17670: 17667: 17664: 17661: 17657: 17654: 17651: 17636: 17633: 17630: 17627: 17624: 17621: 17618: 17615: 17612: 17609: 17606: 17603: 17600: 17597: 17594: 17591: 17588: 17585: 17582: 17579: 17576: 17573: 17570: 17567: 17564: 17561: 17558: 17555: 17551: 17548: 17545: 17541: 17538: 17535: 17532: 17529: 17526: 17523: 17520: 17517: 17514: 17511: 17508: 17505: 17502: 17499: 17496: 17493: 17490: 17487: 17484: 17481: 17478: 17475: 17472: 17469: 17466: 17463: 17460: 17457: 17454: 17451: 17448: 17445: 17442: 17439: 17436: 17433: 17430: 17427: 17424: 17421: 17418: 17415: 17412: 17409: 17406: 17403: 17400: 17397: 17394: 17391: 17388: 17385: 17382: 17379: 17376: 17373: 17370: 17367: 17364: 17360: 17357: 17354: 17350: 17292: 17283:capacity (up to 17275: 17273: 17272: 17267: 17262: 17261: 17216: 17204: 17192: 17115: 17113: 17112: 17107: 17105: 17086: 17083: 17076: 17075: 17060: 17059: 17041: 17040: 17025: 17024: 17012: 17011: 16996: 16995: 16973: 16972: 16956: 16955: 16946: 16945: 16921: 16920: 16905: 16904: 16892: 16891: 16879: 16877: 16876: 16864: 16852: 16851: 16831: 16829: 16828: 16823: 16821: 16820: 16804: 16802: 16801: 16796: 16794: 16793: 16762: 16760: 16759: 16754: 16752: 16751: 16735: 16733: 16732: 16727: 16725: 16724: 16706: 16704: 16703: 16698: 16681: 16678: 16672: 16671: 16653: 16652: 16640: 16639: 16627: 16626: 16595:coefficients of 16276: 16275: 15859: 15857: 15856: 15851: 15849: 15848: 15841: 15840: 15824: 15823: 15810: 15809: 15800: 15799: 15765: 15764: 15755: 15754: 15741: 15740: 15731: 15730: 15717: 15716: 15707: 15706: 15686: 15685: 15678: 15677: 15659: 15658: 15649: 15648: 15636: 15635: 15622: 15621: 15610: 15609: 15600: 15599: 15587: 15586: 15577: 15576: 15564: 15563: 15550: 15549: 15538: 15537: 15528: 15527: 15515: 15514: 15505: 15504: 15492: 15491: 15482: 15481: 15469: 15468: 15455: 15454: 15443: 15442: 15433: 15432: 15420: 15419: 15410: 15409: 15397: 15396: 15387: 15386: 15374: 15373: 15360: 15359: 15348: 15347: 15338: 15337: 15325: 15324: 15315: 15314: 15302: 15301: 15292: 15291: 15279: 15278: 15265: 15264: 15253: 15252: 15243: 15242: 15230: 15229: 15220: 15219: 15207: 15206: 15197: 15196: 15183: 15182: 15171: 15170: 15161: 15160: 15148: 15147: 15138: 15137: 15124: 15123: 15112: 15111: 15102: 15101: 15067: 15065: 15064: 15059: 15042: 15041: 14984: 14982: 14981: 14976: 14974: 14970: 14969: 14954: 14953: 14935: 14934: 14919: 14918: 14900: 14899: 14890: 14889: 14848: 14847: 14829: 14828: 14813: 14812: 14794: 14793: 14784: 14783: 14751: 14750: 14735: 14734: 14716: 14715: 14700: 14699: 14681: 14680: 14665: 14664: 14408: 14407: 14372: 14370: 14369: 14364: 14362: 14361: 14344: 14343: 14342: 14324: 14323: 14306: 14305: 14304: 14292: 14291: 14237: 14235: 14234: 14229: 14227: 14226: 14211: 14210: 14195: 14194: 14185: 14184: 14172: 14171: 14162: 14161: 14105:Forney algorithm 14074: 14072: 14071: 14066: 14064: 14063: 14035: 14034: 14027: 14026: 14013: 14012: 13995: 13994: 13946: 13944: 13943: 13938: 13936: 13935: 13907: 13906: 13872: 13871: 13864: 13863: 13850: 13849: 13832: 13831: 13790: 13788: 13787: 13782: 13771: 13770: 13752: 13751: 13729: 13728: 13710: 13709: 13687: 13686: 13668: 13667: 13652: 13650: 13649: 13644: 13618: 13617: 13599: 13598: 13580: 13579: 13561: 13560: 13542: 13541: 13520: 13519: 13501: 13500: 13476: 13474: 13473: 13468: 13451: 13450: 13435: 13434: 13419: 13418: 13403: 13402: 13387: 13386: 13322: 13320: 13319: 13314: 13297: 13296: 13281: 13280: 13265: 13264: 13249: 13248: 13233: 13232: 13208: 13207: 13195: 13194: 13151: 13149: 13148: 13143: 13126: 13125: 13110: 13109: 13085: 13084: 13075: 13074: 13040: 13039: 13023: 13004: 13002: 13001: 12996: 12979: 12978: 12963: 12962: 12947: 12946: 12931: 12930: 12909: 12908: 12887: 12886: 12827: 12820: 12813: 12742: 12740: 12739: 12734: 12706:Forney algorithm 12673: 12671: 12670: 12665: 12662: 12654: 12638: 12636: 12635: 12630: 12628: 12627: 12569: 12567: 12566: 12561: 12559: 12558: 12551: 12550: 12521: 12520: 12498: 12497: 12468: 12467: 12460: 12459: 12439: 12438: 12419: 12418: 12401: 12400: 12393: 12392: 12367: 12366: 12349: 12348: 12313: 12312: 12290: 12289: 12278: 12277: 12264: 12263: 12247: 12246: 12235: 12234: 12190: 12188: 12187: 12182: 12180: 12179: 12158: 12157: 12148: 12147: 12117: 12116: 12101: 12100: 12082: 12081: 12072: 12071: 12055: 12053: 12052: 12047: 12045: 12044: 12020: 12018: 12017: 12012: 12004: 12003: 11994: 11993: 11981: 11980: 11965: 11964: 11940: 11939: 11918: 11917: 11905: 11904: 11880: 11878: 11877: 11872: 11864: 11860: 11858: 11853: 11844: 11843: 11833: 11828: 11808: 11807: 11789: 11785: 11783: 11766: 11757: 11756: 11746: 11741: 11721: 11720: 11708: 11704: 11702: 11685: 11676: 11675: 11665: 11660: 11640: 11639: 11627: 11623: 11621: 11610: 11601: 11600: 11590: 11585: 11558: 11556: 11555: 11550: 11536: 11534: 11533: 11528: 11520: 11516: 11514: 11509: 11500: 11499: 11490: 11489: 11479: 11474: 11445: 11441: 11439: 11422: 11413: 11412: 11403: 11402: 11392: 11387: 11364: 11360: 11358: 11341: 11332: 11331: 11322: 11321: 11311: 11306: 11283: 11279: 11277: 11266: 11257: 11256: 11246: 11241: 11212: 11210: 11209: 11204: 11196: 11192: 11190: 11185: 11176: 11175: 11166: 11165: 11146: 11129: 11120: 11119: 11110: 11109: 11096: 11079: 11070: 11069: 11060: 11059: 11046: 11035: 11026: 11025: 11010: 11005: 10973: 10971: 10970: 10965: 10963: 10952: 10947: 10938: 10937: 10928: 10927: 10908: 10891: 10882: 10881: 10872: 10871: 10858: 10841: 10832: 10831: 10822: 10821: 10808: 10797: 10788: 10787: 10777: 10766: 10758: 10748: 10737: 10728: 10727: 10718: 10717: 10698: 10690: 10680: 10669: 10660: 10659: 10650: 10649: 10636: 10628: 10618: 10607: 10598: 10597: 10588: 10587: 10574: 10563: 10554: 10553: 10543: 10533: 10529: 10527: 10519: 10510: 10509: 10490: 10482: 10473: 10472: 10459: 10451: 10442: 10441: 10420: 10409: 10400: 10399: 10389: 10375: 10367: 10351: 10340: 10331: 10330: 10320: 10309: 10307: 10306: 10301: 10298: 10287: 10278: 10277: 10261: 10259: 10258: 10253: 10229: 10227: 10226: 10221: 10207: 10205: 10204: 10199: 10187: 10179: 10157: 10155: 10154: 10149: 10126: 10125: 10112: 10104: 10079: 10077: 10076: 10071: 10068: 10060: 10038: 10036: 10035: 10030: 10027: 10019: 10003: 9993: 9991: 9990: 9985: 9983: 9982: 9973: 9972: 9954: 9953: 9944: 9943: 9931: 9930: 9921: 9920: 9899: 9898: 9876: 9871: 9834: 9805: 9803: 9802: 9797: 9785: 9783: 9782: 9777: 9759: 9757: 9756: 9751: 9705: 9703: 9702: 9697: 9695: 9694: 9687: 9686: 9660: 9659: 9646: 9645: 9625: 9624: 9617: 9616: 9596: 9595: 9582: 9581: 9564: 9563: 9555: 9544: 9527: 9516: 9504: 9493: 9457: 9452: 9435: 9430: 9418: 9413: 9399: 9394: 9377: 9372: 9360: 9355: 9306: 9289: 9287: 9286: 9281: 9278: 9273: 9261: 9260: 9251: 9250: 9249: 9248: 9228: 9227: 9226: 9225: 9202: 9201: 9200: 9199: 9185: 9184: 9163: 9161: 9160: 9155: 9152: 9147: 9138: 9137: 9127: 9122: 9104: 9103: 9085: 9083: 9082: 9077: 9075: 9074: 9073: 9072: 9055: 9054: 9043: 9039: 9038: 9037: 9036: 9019: 9018: 8970: 8968: 8967: 8962: 8960: 8959: 8943: 8941: 8940: 8935: 8914: 8912: 8911: 8906: 8895: 8894: 8872: 8870: 8869: 8864: 8862: 8821: 8820: 8819: 8818: 8808: 8804: 8803: 8789: 8788: 8787: 8786: 8771: 8766: 8745: 8738: 8737: 8716: 8709: 8708: 8681: 8680: 8659: 8658: 8637: 8636: 8614: 8613: 8584: 8582: 8581: 8576: 8574: 8573: 8555: 8554: 8483: 8481: 8480: 8475: 8473: 8472: 8471: 8470: 8456: 8455: 8454: 8453: 8438: 8433: 8368: 8366: 8365: 8360: 8358: 8357: 8348: 8347: 8337: 8326: 8290: 8288: 8287: 8282: 8213: 8211: 8210: 8205: 8168: 8155: 8154: 8132: 8130: 8129: 8124: 8113: 8112: 8097: 8096: 8069: 8068: 8053: 8052: 8001: 7999: 7998: 7993: 7985: 7984: 7965: 7954: 7899: 7897: 7896: 7891: 7889: 7888: 7879: 7878: 7868: 7857: 7812: 7810: 7809: 7804: 7799: 7798: 7774: 7773: 7755: 7754: 7668: 7666: 7665: 7660: 7644: 7642: 7641: 7636: 7634: 7633: 7621: 7620: 7604: 7602: 7601: 7596: 7588: 7587: 7560: 7559: 7534: 7532: 7531: 7526: 7496: 7494: 7493: 7488: 7480: 7479: 7446: 7445: 7420: 7418: 7417: 7412: 7400: 7398: 7397: 7392: 7380: 7378: 7377: 7372: 7370: 7362: 7344: 7342: 7341: 7336: 7294: 7292: 7291: 7286: 7284: 7283: 7267: 7265: 7264: 7259: 7247: 7245: 7244: 7239: 7223: 7221: 7220: 7215: 7173: 7171: 7170: 7165: 7123: 7121: 7120: 7115: 7097: 7095: 7094: 7089: 7067: 7065: 7064: 7059: 7041: 7039: 7038: 7033: 7017: 7015: 7014: 7009: 6995: 6994: 6972: 6970: 6969: 6964: 6956: 6955: 6933: 6931: 6930: 6925: 6913: 6911: 6910: 6905: 6903: 6902: 6886: 6884: 6883: 6878: 6863: 6861: 6860: 6855: 6843: 6841: 6840: 6835: 6823: 6821: 6820: 6815: 6813: 6811: 6804: 6803: 6790: 6775: 6773: 6772: 6767: 6765: 6764: 6734: 6733: 6717: 6715: 6714: 6709: 6698: 6697: 6685: 6677: 6662: 6660: 6659: 6654: 6652: 6651: 6636: 6635: 6616: 6611: 6602: 6601: 6600: 6587: 6576: 6571: 6550: 6542: 6540: 6532: 6527: 6526: 6510: 6508: 6507: 6502: 6500: 6499: 6484: 6483: 6464: 6459: 6450: 6449: 6448: 6435: 6424: 6419: 6398: 6390: 6388: 6386: 6379: 6378: 6368: 6367: 6352: 6347: 6346: 6311: 6309: 6308: 6303: 6291: 6289: 6288: 6283: 6271: 6237: 6235: 6234: 6229: 6224: 6197: 6195: 6194: 6189: 6152: 6150: 6149: 6144: 6124: 6123: 6107:Hamming distance 6078: 6076: 6075: 6070: 6031: 6030: 6009: 6008: 5987: 5986: 5974: 5973: 5927: 5925: 5924: 5919: 5898: 5896: 5895: 5890: 5888: 5876: 5874: 5873: 5868: 5845: 5843: 5842: 5837: 5822: 5821: 5809: 5808: 5762: 5760: 5759: 5754: 5733: 5731: 5730: 5725: 5713: 5711: 5710: 5705: 5684: 5682: 5681: 5676: 5674: 5673: 5642: 5640: 5639: 5634: 5632: 5631: 5619: 5618: 5600: 5599: 5581: 5580: 5558: 5556: 5555: 5550: 5530: 5528: 5527: 5522: 5505: 5504: 5501: 5493: 5492: 5491: 5455: 5454: 5438: 5436: 5435: 5430: 5419: 5418: 5402: 5400: 5399: 5394: 5382: 5380: 5379: 5374: 5353: 5351: 5350: 5345: 5321: 5319: 5318: 5313: 5311: 5310: 5294: 5292: 5291: 5286: 5262: 5260: 5259: 5254: 5233: 5231: 5230: 5225: 5204: 5202: 5201: 5196: 5175: 5173: 5172: 5167: 5146: 5144: 5143: 5138: 5117: 5115: 5114: 5109: 5088: 5086: 5085: 5080: 5059: 5057: 5056: 5051: 5039: 5037: 5036: 5031: 5010: 5008: 5007: 5002: 4943: 4941: 4940: 4935: 4930: 4926: 4925: 4924: 4900: 4899: 4887: 4886: 4877: 4874: 4872: 4871: 4862: 4861: 4851: 4846: 4815: 4814: 4807: 4803: 4802: 4801: 4783: 4782: 4770: 4769: 4747: 4733: 4731: 4730: 4725: 4705: 4703: 4702: 4697: 4695: 4694: 4676: 4675: 4654: 4653: 4620: 4619: 4607: 4606: 4594: 4590: 4589: 4588: 4547: 4543: 4542: 4541: 4515: 4511: 4510: 4509: 4468: 4466: 4465: 4460: 4448: 4446: 4445: 4440: 4419: 4417: 4416: 4411: 4393: 4391: 4390: 4385: 4361: 4359: 4358: 4353: 4335: 4333: 4332: 4327: 4306: 4304: 4303: 4298: 4277: 4275: 4274: 4269: 4248: 4246: 4245: 4240: 4216: 4214: 4213: 4208: 4190: 4188: 4187: 4182: 4161: 4159: 4158: 4153: 4117: 4115: 4114: 4109: 4104: 4103: 4079: 4078: 4051: 4050: 4026: 4025: 4010: 4008: 4007: 4002: 4000: 3999: 3987: 3986: 3952: 3950: 3949: 3944: 3942: 3941: 3931: 3930: 3912: 3911: 3888: 3887: 3875: 3874: 3858: 3857: 3845: 3844: 3805: 3803: 3802: 3797: 3795: 3794: 3766: 3764: 3763: 3758: 3756: 3755: 3739: 3737: 3736: 3731: 3729: 3728: 3704: 3703: 3687: 3685: 3684: 3679: 3674: 3673: 3655: 3654: 3633: 3632: 3620: 3619: 3603: 3601: 3600: 3595: 3584:since the first 3580:This variant is 3577: 3575: 3574: 3569: 3567: 3566: 3556: 3555: 3537: 3536: 3513: 3512: 3500: 3499: 3483: 3482: 3470: 3469: 3428: 3426: 3425: 3420: 3418: 3417: 3393: 3392: 3372: 3370: 3369: 3364: 3352: 3350: 3349: 3344: 3342: 3341: 3325: 3323: 3322: 3317: 3279: 3276: 3274: 3273: 3258: 3257: 3245: 3244: 3228: 3226: 3225: 3220: 3208: 3206: 3205: 3200: 3198: 3197: 3181: 3179: 3178: 3173: 3171: 3170: 3142: 3140: 3139: 3134: 3121:generator matrix 3114: 3112: 3111: 3106: 3088: 3086: 3085: 3080: 3078: 3077: 3070: 3069: 3043: 3042: 3029: 3028: 3011: 3010: 3002: 2991: 2968: 2963: 2946: 2945: 2893: 2882: 2865: 2860: 2849: 2848: 2829: 2818: 2801: 2796: 2785: 2784: 2731: 2729: 2728: 2723: 2711: 2709: 2708: 2703: 2691: 2689: 2688: 2683: 2665: 2663: 2662: 2657: 2623: 2621: 2620: 2615: 2603: 2601: 2600: 2595: 2593: 2592: 2582: 2581: 2563: 2562: 2539: 2538: 2526: 2525: 2509: 2508: 2496: 2495: 2456: 2454: 2453: 2448: 2446: 2445: 2433: 2432: 2410: 2408: 2407: 2402: 2390: 2388: 2387: 2382: 2380: 2379: 2355: 2354: 2338: 2336: 2335: 2330: 2318: 2316: 2315: 2310: 2308: 2307: 2291: 2289: 2288: 2283: 2272:The codeword of 2271: 2269: 2268: 2263: 2257: 2256: 2247: 2246: 2236: 2225: 2198: 2197: 2181: 2179: 2178: 2173: 2171: 2170: 2154: 2152: 2151: 2146: 2144: 2143: 2128: 2127: 2103: 2102: 2053: 2051: 2050: 2045: 2043: 2042: 2024: 2023: 1983: 1981: 1980: 1975: 1973: 1972: 1945: 1943: 1942: 1937: 1932: 1891: 1889: 1888: 1883: 1878: 1857: 1855: 1854: 1849: 1832: 1806: 1792: 1772: 1751: 1749: 1748: 1743: 1709: 1707: 1706: 1701: 1683: 1681: 1680: 1675: 1627: 1625: 1624: 1619: 1601: 1599: 1598: 1593: 1577: 1575: 1574: 1569: 1563: 1562: 1549: 1546: 1541: 1538: 1532: 1531: 1524: 1523: 1514: 1513: 1486: 1485: 1464: 1463: 1448: 1447: 1440: 1439: 1430: 1418: 1416: 1415: 1410: 1408: 1341: 1339: 1338: 1333: 1331: 1330: 1312: 1311: 1296:distinct points 1288:is evaluated at 1283: 1281: 1280: 1275: 1263: 1261: 1260: 1255: 1239: 1237: 1236: 1231: 1219: 1217: 1216: 1211: 1187: 1185: 1184: 1179: 1155: 1153: 1152: 1147: 1129: 1127: 1126: 1121: 1119: 1111: 1089: 1087: 1086: 1081: 1069: 1067: 1066: 1061: 1049: 1047: 1046: 1041: 1026: 1024: 1023: 1018: 991: 989: 988: 983: 968: 966: 965: 960: 945: 943: 942: 937: 828:Mariner, Viking 823:Reed-Muller code 780: 776: 765:Shannon capacity 738:Viterbi decoders 639:erasure channels 354: 351:. The choice of 350: 346: 335: 327: 315: 311: 307: 303: 216: 209: 202: 170:Berlekamp–Massey 30: 29: 21: 22584: 22583: 22579: 22578: 22577: 22575: 22574: 22573: 22554: 22553: 22515: 22513:Implementations 22505: 22444: 22439: 22430: 22414:Reed, Irving S. 22407: 22350:Forney, Jr., G. 22323: 22283:, Lecture Notes 22280: 22268:10.1137/0108018 22248:Reed, Irving S. 22195: 22182: 22147: 22134: 22132: 22128: 22121: 22111: 22109:Further reading 22106: 22105: 22096: 22094: 22090: 22079: 22073: 22069: 22034: 22030: 22022: 22020: 22002: 21998: 21991: 21955: 21951: 21944: 21908: 21904: 21870: 21866: 21861: 21857: 21849: 21845: 21840: 21836: 21823: 21821: 21817: 21802: 21796: 21792: 21783: 21781: 21772: 21771: 21767: 21758: 21754: 21733:(11): 2142–56. 21722: 21716: 21709: 21694: 21680: 21676: 21670: 21649: 21645: 21637: 21629: 21622: 21593: 21589: 21574: 21558: 21554: 21535: 21531: 21516:10.1137/0109020 21500: 21496: 21489: 21485: 21480: 21475: 21474: 21465: 21461: 21456: 21414: 21409: 21403: 21395: 21384: 21373: 21370: 21336: 21334: 21316: 21314: 21288: 21259: 21248: 21234: 21232: 21213: 21199: 21149: 21141: 21103: 21099: 21097: 21094: 21093: 21066: 21062: 21060: 21057: 21056: 21024: 21020: 21005: 21001: 20996: 20993: 20992: 20972: 20968: 20966: 20963: 20962: 20941: 20937: 20922: 20911: 20895: 20891: 20889: 20886: 20885: 20879: 20871: 20866: 20860: 20852: 20848: 20844: 20841: 20807: 20805: 20787: 20785: 20759: 20740: 20739: 20733: 20732: 20726: 20725: 20719: 20718: 20712: 20711: 20705: 20704: 20698: 20697: 20687: 20686: 20676: 20675: 20669: 20665: 20662: 20661: 20655: 20651: 20648: 20647: 20641: 20637: 20634: 20633: 20627: 20623: 20620: 20619: 20613: 20609: 20606: 20605: 20599: 20595: 20592: 20591: 20585: 20581: 20574: 20573: 20566: 20565: 20560: 20555: 20550: 20545: 20540: 20535: 20529: 20528: 20523: 20518: 20513: 20508: 20503: 20498: 20492: 20491: 20486: 20481: 20476: 20471: 20466: 20461: 20455: 20454: 20449: 20444: 20439: 20434: 20429: 20424: 20418: 20417: 20412: 20407: 20402: 20397: 20392: 20387: 20381: 20380: 20375: 20370: 20365: 20360: 20355: 20350: 20344: 20343: 20338: 20333: 20328: 20323: 20318: 20313: 20303: 20302: 20300: 20297: 20296: 20270: 20269: 20263: 20262: 20256: 20255: 20249: 20248: 20242: 20241: 20235: 20234: 20228: 20227: 20217: 20216: 20206: 20205: 20199: 20195: 20192: 20191: 20185: 20181: 20178: 20177: 20171: 20167: 20164: 20163: 20157: 20153: 20150: 20149: 20143: 20139: 20136: 20135: 20129: 20125: 20122: 20121: 20115: 20111: 20104: 20103: 20096: 20095: 20090: 20085: 20080: 20075: 20070: 20065: 20059: 20058: 20053: 20048: 20043: 20038: 20033: 20028: 20022: 20021: 20016: 20011: 20006: 20001: 19996: 19991: 19985: 19984: 19979: 19974: 19969: 19964: 19959: 19954: 19948: 19947: 19942: 19937: 19932: 19927: 19922: 19917: 19911: 19910: 19905: 19900: 19895: 19890: 19885: 19880: 19874: 19873: 19868: 19863: 19858: 19853: 19848: 19843: 19833: 19832: 19830: 19827: 19826: 19808: 19803: 19793: 19789: 19774: 19769: 19759: 19755: 19746: 19741: 19731: 19727: 19718: 19713: 19703: 19699: 19690: 19686: 19680: 19676: 19667: 19663: 19648: 19644: 19638: 19634: 19625: 19621: 19612: 19608: 19606: 19603: 19602: 19568: 19564: 19546: 19542: 19530: 19526: 19524: 19521: 19520: 19515: 19501: 19496: 19490: 19485: 19467: 19462: 19456: 19448: 19438: 19417: 19414: 19413: 19393: 19389: 19381: 19378: 19377: 19369: 19336: 19333: 19332: 19293: 19285: 19283: 19273: 19260: 19259: 19258: 19256: 19253: 19252: 19230: 19226: 19224: 19221: 19220: 19203: 19199: 19197: 19194: 19193: 19187: 19178: 19173: 19172: 19169: 19166: 19163: 19160: 19157: 19154: 19151: 19148: 19145: 19142: 19139: 19136: 19133: 19130: 19127: 19124: 19120: 19117: 19114: 19110: 19107: 19104: 19101: 19098: 19095: 19092: 19089: 19086: 19083: 19080: 19077: 19074: 19071: 19068: 19065: 19062: 19059: 19056: 19053: 19050: 19047: 19044: 19041: 19038: 19035: 19032: 19029: 19026: 19023: 19020: 19017: 19014: 19010: 19007: 19004: 19000: 18997: 18994: 18991: 18988: 18985: 18982: 18979: 18976: 18973: 18970: 18967: 18964: 18961: 18958: 18955: 18952: 18949: 18946: 18943: 18940: 18937: 18934: 18931: 18928: 18925: 18922: 18919: 18916: 18913: 18910: 18907: 18904: 18901: 18898: 18895: 18892: 18889: 18886: 18883: 18880: 18877: 18874: 18871: 18868: 18865: 18862: 18859: 18856: 18853: 18850: 18847: 18844: 18841: 18838: 18835: 18832: 18829: 18826: 18823: 18820: 18817: 18814: 18811: 18808: 18805: 18802: 18799: 18796: 18793: 18790: 18787: 18784: 18781: 18778: 18775: 18772: 18769: 18766: 18763: 18760: 18757: 18754: 18751: 18748: 18745: 18742: 18739: 18736: 18733: 18730: 18727: 18724: 18721: 18718: 18715: 18712: 18709: 18706: 18703: 18700: 18697: 18694: 18691: 18688: 18685: 18682: 18679: 18676: 18673: 18670: 18667: 18664: 18661: 18658: 18655: 18652: 18649: 18646: 18643: 18640: 18637: 18634: 18631: 18628: 18625: 18622: 18619: 18616: 18613: 18610: 18607: 18604: 18601: 18598: 18595: 18592: 18589: 18586: 18583: 18580: 18577: 18574: 18571: 18568: 18565: 18562: 18559: 18556: 18553: 18550: 18547: 18544: 18541: 18538: 18535: 18532: 18529: 18526: 18523: 18520: 18517: 18514: 18511: 18508: 18505: 18502: 18499: 18496: 18493: 18490: 18487: 18484: 18481: 18478: 18475: 18472: 18469: 18466: 18463: 18460: 18457: 18454: 18451: 18448: 18445: 18442: 18439: 18436: 18433: 18430: 18427: 18424: 18421: 18418: 18415: 18412: 18409: 18406: 18403: 18400: 18397: 18394: 18391: 18388: 18385: 18382: 18379: 18376: 18373: 18370: 18367: 18364: 18361: 18358: 18355: 18352: 18349: 18346: 18343: 18340: 18337: 18334: 18331: 18328: 18325: 18322: 18319: 18316: 18313: 18310: 18307: 18304: 18301: 18298: 18295: 18292: 18289: 18286: 18283: 18280: 18277: 18274: 18271: 18268: 18265: 18262: 18259: 18256: 18253: 18250: 18247: 18244: 18241: 18238: 18235: 18232: 18229: 18226: 18223: 18220: 18217: 18214: 18211: 18208: 18205: 18202: 18199: 18196: 18193: 18190: 18187: 18184: 18181: 18178: 18175: 18172: 18169: 18166: 18163: 18160: 18157: 18154: 18151: 18148: 18145: 18142: 18139: 18136: 18133: 18130: 18127: 18124: 18121: 18118: 18115: 18112: 18109: 18106: 18103: 18100: 18097: 18094: 18091: 18088: 18085: 18082: 18079: 18076: 18073: 18070: 18067: 18064: 18061: 18058: 18055: 18052: 18049: 18046: 18043: 18040: 18037: 18034: 18031: 18028: 18025: 18022: 18019: 18016: 18013: 18010: 18007: 18004: 18001: 17998: 17995: 17992: 17989: 17986: 17983: 17980: 17977: 17974: 17971: 17968: 17965: 17962: 17959: 17956: 17953: 17950: 17947: 17944: 17941: 17938: 17935: 17932: 17929: 17926: 17923: 17920: 17917: 17914: 17911: 17908: 17905: 17902: 17899: 17896: 17893: 17890: 17887: 17884: 17881: 17878: 17875: 17872: 17869: 17866: 17863: 17860: 17857: 17854: 17851: 17848: 17845: 17842: 17839: 17836: 17833: 17830: 17827: 17824: 17821: 17818: 17815: 17812: 17809: 17806: 17803: 17800: 17797: 17794: 17791: 17788: 17785: 17782: 17779: 17776: 17773: 17770: 17768:% Get the alpha 17767: 17764: 17761: 17758: 17755: 17752: 17749: 17746: 17743: 17740: 17737: 17734: 17731: 17728: 17725: 17722: 17719: 17716: 17713: 17710: 17707: 17704: 17701: 17698: 17695: 17692: 17689: 17686: 17683: 17680: 17677: 17674: 17671: 17668: 17665: 17662: 17659: 17655: 17652: 17649: 17643: 17638: 17637: 17634: 17631: 17628: 17625: 17622: 17619: 17616: 17613: 17610: 17607: 17604: 17601: 17598: 17595: 17592: 17589: 17586: 17583: 17580: 17577: 17574: 17571: 17568: 17565: 17562: 17559: 17556: 17553: 17549: 17546: 17543: 17539: 17536: 17533: 17530: 17527: 17524: 17521: 17518: 17515: 17512: 17509: 17506: 17503: 17500: 17497: 17494: 17491: 17488: 17485: 17482: 17479: 17476: 17473: 17470: 17467: 17464: 17461: 17458: 17455: 17452: 17449: 17446: 17443: 17440: 17437: 17434: 17431: 17428: 17425: 17422: 17419: 17416: 17413: 17410: 17407: 17404: 17401: 17398: 17395: 17392: 17389: 17387:% Get the alpha 17386: 17383: 17380: 17377: 17374: 17371: 17368: 17365: 17362: 17358: 17355: 17352: 17348: 17338: 17333: 17324:Alexander Vardy 17299: 17284: 17257: 17253: 17242: 17239: 17238: 17206: 17198: 17193:. The distance 17183: 17168:Singleton bound 17164: 17118:Then calculate 17103: 17102: 17082: 17080: 17065: 17061: 17055: 17051: 17030: 17026: 17020: 17016: 17001: 16997: 16991: 16987: 16974: 16968: 16964: 16961: 16960: 16951: 16947: 16935: 16931: 16910: 16906: 16900: 16896: 16887: 16883: 16872: 16868: 16863: 16853: 16847: 16843: 16839: 16837: 16834: 16833: 16816: 16812: 16810: 16807: 16806: 16789: 16785: 16783: 16780: 16779: 16747: 16743: 16741: 16738: 16737: 16720: 16716: 16714: 16711: 16710: 16677: 16667: 16663: 16648: 16644: 16635: 16631: 16622: 16618: 16616: 16613: 16612: 16442: 16437: 16431: 16419: 16409: 16297: 16289: 16271: 16264: 16260: 16251: 16249: 16240: 16226: 16224: 16215: 16199: 16177: 16176: 16163: 16153: 16145: 16132: 16122: 16114: 16104: 16078: 16061: 16054: 16039: 16029: 16004: 15994: 15985: 15975: 15966: 15956: 15937: 15919: 15913: 15896: 15875: 15843: 15842: 15836: 15832: 15829: 15828: 15819: 15815: 15812: 15811: 15805: 15801: 15795: 15791: 15788: 15787: 15781: 15780: 15774: 15773: 15767: 15766: 15760: 15756: 15750: 15746: 15743: 15742: 15736: 15732: 15726: 15722: 15719: 15718: 15712: 15708: 15702: 15698: 15691: 15690: 15680: 15679: 15673: 15669: 15666: 15665: 15660: 15654: 15650: 15644: 15640: 15631: 15627: 15624: 15623: 15617: 15613: 15611: 15605: 15601: 15595: 15591: 15582: 15578: 15572: 15568: 15559: 15555: 15552: 15551: 15545: 15541: 15539: 15533: 15529: 15523: 15519: 15510: 15506: 15500: 15496: 15487: 15483: 15477: 15473: 15464: 15460: 15457: 15456: 15450: 15446: 15444: 15438: 15434: 15428: 15424: 15415: 15411: 15405: 15401: 15392: 15388: 15382: 15378: 15369: 15365: 15362: 15361: 15355: 15351: 15349: 15343: 15339: 15333: 15329: 15320: 15316: 15310: 15306: 15297: 15293: 15287: 15283: 15274: 15270: 15267: 15266: 15260: 15256: 15254: 15248: 15244: 15238: 15234: 15225: 15221: 15215: 15211: 15202: 15198: 15192: 15188: 15185: 15184: 15178: 15174: 15172: 15166: 15162: 15156: 15152: 15143: 15139: 15133: 15129: 15126: 15125: 15119: 15115: 15113: 15107: 15103: 15097: 15093: 15086: 15085: 15083: 15080: 15079: 15037: 15033: 14992: 14989: 14988: 14972: 14971: 14965: 14961: 14949: 14945: 14924: 14920: 14908: 14904: 14895: 14891: 14885: 14881: 14874: 14859: 14858: 14843: 14839: 14818: 14814: 14802: 14798: 14789: 14785: 14779: 14775: 14768: 14753: 14752: 14746: 14742: 14730: 14726: 14705: 14701: 14689: 14685: 14670: 14666: 14660: 14656: 14649: 14633: 14631: 14628: 14627: 14593: 14424: 14403: 14351: 14347: 14338: 14334: 14313: 14309: 14300: 14296: 14287: 14283: 14275: 14272: 14271: 14256: 14222: 14218: 14206: 14202: 14190: 14186: 14180: 14176: 14167: 14163: 14157: 14153: 14151: 14148: 14147: 14144: 14142: 14138: 14134: 14129: 14127: 14123: 14119: 14114: 14111: 14098: 14092: 14090: 14083: 14058: 14057: 14051: 14050: 14040: 14039: 14029: 14028: 14022: 14018: 14015: 14014: 14008: 14004: 13997: 13996: 13989: 13988: 13983: 13977: 13976: 13971: 13961: 13960: 13958: 13955: 13954: 13930: 13929: 13923: 13922: 13912: 13911: 13901: 13900: 13891: 13890: 13877: 13876: 13866: 13865: 13859: 13855: 13852: 13851: 13845: 13841: 13834: 13833: 13826: 13825: 13820: 13814: 13813: 13808: 13798: 13797: 13795: 13792: 13791: 13766: 13762: 13747: 13743: 13724: 13720: 13705: 13701: 13682: 13678: 13663: 13659: 13657: 13654: 13653: 13613: 13609: 13594: 13590: 13575: 13571: 13556: 13552: 13537: 13533: 13515: 13511: 13496: 13492: 13490: 13487: 13486: 13446: 13442: 13430: 13426: 13414: 13410: 13398: 13394: 13382: 13378: 13328: 13325: 13324: 13292: 13288: 13276: 13272: 13260: 13256: 13244: 13240: 13228: 13224: 13203: 13199: 13190: 13186: 13156: 13153: 13152: 13121: 13117: 13105: 13101: 13080: 13076: 13070: 13066: 13035: 13031: 13029: 13026: 13025: 13006: 12974: 12970: 12958: 12954: 12942: 12938: 12926: 12922: 12904: 12900: 12882: 12878: 12837: 12834: 12833: 12822: 12815: 12808: 12805: 12781: 12780: 12771: 12756: 12748: 12728: 12725: 12724: 12721: 12714: 12702:error equations 12698: 12691: 12684: 12655: 12650: 12644: 12641: 12640: 12623: 12619: 12617: 12614: 12613: 12610: 12604: 12597: 12553: 12552: 12540: 12536: 12530: 12529: 12523: 12522: 12510: 12506: 12500: 12499: 12487: 12483: 12473: 12472: 12462: 12461: 12455: 12451: 12448: 12447: 12441: 12440: 12428: 12424: 12421: 12420: 12414: 12410: 12403: 12402: 12395: 12394: 12379: 12375: 12373: 12368: 12356: 12352: 12350: 12344: 12340: 12337: 12336: 12331: 12326: 12321: 12315: 12314: 12302: 12298: 12296: 12291: 12285: 12281: 12279: 12273: 12269: 12266: 12265: 12259: 12255: 12253: 12248: 12242: 12238: 12236: 12230: 12226: 12219: 12218: 12216: 12213: 12212: 12210: 12169: 12165: 12153: 12149: 12131: 12127: 12106: 12102: 12090: 12086: 12077: 12073: 12067: 12063: 12061: 12058: 12057: 12034: 12030: 12028: 12025: 12024: 11999: 11995: 11989: 11985: 11970: 11966: 11954: 11950: 11923: 11919: 11913: 11909: 11894: 11890: 11888: 11885: 11884: 11854: 11849: 11839: 11835: 11829: 11818: 11813: 11809: 11803: 11799: 11767: 11762: 11752: 11748: 11742: 11731: 11726: 11722: 11716: 11712: 11686: 11681: 11671: 11667: 11661: 11650: 11645: 11641: 11635: 11631: 11611: 11606: 11596: 11592: 11586: 11575: 11570: 11566: 11564: 11561: 11560: 11544: 11541: 11540: 11510: 11505: 11495: 11491: 11485: 11481: 11475: 11464: 11459: 11455: 11423: 11418: 11408: 11404: 11398: 11394: 11388: 11377: 11372: 11368: 11342: 11337: 11327: 11323: 11317: 11313: 11307: 11296: 11291: 11287: 11267: 11262: 11252: 11248: 11242: 11231: 11226: 11222: 11220: 11217: 11216: 11186: 11181: 11171: 11167: 11161: 11157: 11130: 11125: 11115: 11111: 11105: 11101: 11080: 11075: 11065: 11061: 11055: 11051: 11036: 11031: 11021: 11017: 11016: 11012: 11006: 10995: 10989: 10986: 10985: 10961: 10960: 10948: 10943: 10933: 10929: 10923: 10919: 10892: 10887: 10877: 10873: 10867: 10863: 10842: 10837: 10827: 10823: 10817: 10813: 10798: 10793: 10783: 10779: 10775: 10774: 10759: 10754: 10738: 10733: 10723: 10719: 10713: 10709: 10691: 10686: 10670: 10665: 10655: 10651: 10645: 10641: 10629: 10624: 10608: 10603: 10593: 10589: 10583: 10579: 10564: 10559: 10549: 10545: 10541: 10540: 10520: 10515: 10505: 10501: 10483: 10478: 10468: 10464: 10452: 10447: 10437: 10433: 10426: 10422: 10410: 10405: 10395: 10391: 10387: 10386: 10368: 10363: 10341: 10336: 10326: 10322: 10317: 10315: 10312: 10311: 10288: 10283: 10273: 10269: 10267: 10264: 10263: 10235: 10232: 10231: 10215: 10212: 10211: 10180: 10175: 10163: 10160: 10159: 10121: 10117: 10105: 10100: 10085: 10082: 10081: 10061: 10056: 10044: 10041: 10040: 10020: 10015: 10009: 10006: 10005: 9997: 9978: 9974: 9968: 9964: 9949: 9945: 9939: 9935: 9926: 9922: 9916: 9912: 9894: 9890: 9872: 9861: 9840: 9837: 9836: 9828: 9819: 9812: 9791: 9788: 9787: 9768: 9765: 9764: 9739: 9736: 9735: 9732: 9719: 9713: 9689: 9688: 9676: 9672: 9669: 9668: 9662: 9661: 9655: 9651: 9648: 9647: 9641: 9637: 9630: 9629: 9619: 9618: 9612: 9608: 9605: 9604: 9598: 9597: 9591: 9587: 9584: 9583: 9577: 9573: 9566: 9565: 9558: 9557: 9545: 9540: 9534: 9529: 9517: 9512: 9506: 9494: 9489: 9482: 9481: 9476: 9471: 9466: 9460: 9459: 9453: 9448: 9442: 9437: 9431: 9426: 9420: 9414: 9409: 9402: 9401: 9395: 9390: 9384: 9379: 9373: 9368: 9362: 9356: 9351: 9340: 9339: 9337: 9334: 9333: 9330: 9323: 9316: 9294: 9274: 9269: 9256: 9252: 9244: 9240: 9239: 9235: 9221: 9217: 9210: 9206: 9195: 9191: 9190: 9186: 9180: 9176: 9171: 9168: 9167: 9148: 9143: 9133: 9129: 9123: 9112: 9099: 9095: 9093: 9090: 9089: 9068: 9064: 9063: 9059: 9050: 9046: 9032: 9028: 9027: 9023: 9014: 9010: 9008: 9005: 9004: 9001: 8991: 8981: 8955: 8951: 8949: 8946: 8945: 8920: 8917: 8916: 8890: 8886: 8878: 8875: 8874: 8860: 8859: 8814: 8810: 8809: 8799: 8795: 8791: 8790: 8782: 8778: 8777: 8773: 8767: 8756: 8743: 8742: 8733: 8729: 8714: 8713: 8704: 8700: 8676: 8672: 8654: 8650: 8632: 8628: 8615: 8609: 8605: 8601: 8599: 8596: 8595: 8593: 8563: 8559: 8550: 8546: 8544: 8541: 8540: 8537: 8507:). From those, 8505: 8504: 8495: 8466: 8462: 8461: 8457: 8449: 8445: 8444: 8440: 8434: 8423: 8402: 8399: 8398: 8391: 8376: 8353: 8349: 8343: 8339: 8327: 8316: 8295: 8292: 8291: 8237: 8234: 8233: 8150: 8146: 8138: 8135: 8134: 8108: 8104: 8092: 8088: 8064: 8060: 8048: 8044: 8030: 8027: 8026: 7980: 7976: 7955: 7944: 7923: 7920: 7919: 7884: 7880: 7874: 7870: 7858: 7847: 7826: 7823: 7822: 7788: 7784: 7769: 7765: 7750: 7746: 7741: 7738: 7737: 7734: 7717: 7711: 7703: 7683: 7654: 7651: 7650: 7629: 7625: 7616: 7612: 7610: 7607: 7606: 7583: 7579: 7555: 7551: 7540: 7537: 7536: 7502: 7499: 7498: 7469: 7465: 7441: 7437: 7426: 7423: 7422: 7406: 7403: 7402: 7386: 7383: 7382: 7366: 7358: 7350: 7347: 7346: 7300: 7297: 7296: 7279: 7275: 7273: 7270: 7269: 7253: 7250: 7249: 7233: 7230: 7229: 7209: 7206: 7205: 7129: 7126: 7125: 7103: 7100: 7099: 7077: 7074: 7073: 7047: 7044: 7043: 7027: 7024: 7023: 6990: 6986: 6978: 6975: 6974: 6951: 6947: 6939: 6936: 6935: 6919: 6916: 6915: 6898: 6894: 6892: 6889: 6888: 6872: 6869: 6868: 6849: 6846: 6845: 6829: 6826: 6825: 6799: 6795: 6794: 6789: 6781: 6778: 6777: 6760: 6756: 6729: 6725: 6723: 6720: 6719: 6693: 6689: 6676: 6668: 6665: 6664: 6641: 6637: 6631: 6627: 6612: 6607: 6596: 6583: 6582: 6581: 6572: 6555: 6541: 6531: 6522: 6518: 6516: 6513: 6512: 6489: 6485: 6479: 6475: 6460: 6455: 6444: 6431: 6430: 6429: 6420: 6403: 6389: 6374: 6370: 6369: 6357: 6353: 6351: 6342: 6338: 6336: 6333: 6332: 6297: 6294: 6293: 6277: 6274: 6273: 6254: 6220: 6203: 6200: 6199: 6177: 6174: 6173: 6163:Singleton bound 6119: 6115: 6113: 6110: 6109: 6084: 6026: 6022: 6004: 6000: 5982: 5978: 5969: 5965: 5933: 5930: 5929: 5904: 5901: 5900: 5884: 5882: 5879: 5878: 5853: 5850: 5849: 5817: 5813: 5804: 5800: 5768: 5765: 5764: 5739: 5736: 5735: 5719: 5716: 5715: 5690: 5687: 5686: 5669: 5665: 5648: 5645: 5644: 5627: 5623: 5614: 5610: 5589: 5585: 5570: 5566: 5564: 5561: 5560: 5538: 5535: 5534: 5500: 5496: 5487: 5483: 5450: 5446: 5444: 5441: 5440: 5414: 5410: 5408: 5405: 5404: 5388: 5385: 5384: 5359: 5356: 5355: 5327: 5324: 5323: 5306: 5302: 5300: 5297: 5296: 5271: 5268: 5267: 5239: 5236: 5235: 5210: 5207: 5206: 5181: 5178: 5177: 5152: 5149: 5148: 5123: 5120: 5119: 5094: 5091: 5090: 5065: 5062: 5061: 5045: 5042: 5041: 5016: 5013: 5012: 4960: 4957: 4956: 4949: 4914: 4910: 4895: 4891: 4882: 4878: 4873: 4867: 4863: 4857: 4853: 4847: 4836: 4810: 4809: 4797: 4793: 4778: 4774: 4765: 4761: 4760: 4756: 4755: 4751: 4743: 4741: 4738: 4737: 4713: 4710: 4709: 4684: 4680: 4659: 4655: 4637: 4633: 4615: 4611: 4602: 4598: 4566: 4562: 4555: 4551: 4531: 4527: 4520: 4516: 4505: 4501: 4494: 4490: 4473: 4470: 4469: 4454: 4451: 4450: 4425: 4422: 4421: 4399: 4396: 4395: 4370: 4367: 4366: 4341: 4338: 4337: 4312: 4309: 4308: 4283: 4280: 4279: 4254: 4251: 4250: 4222: 4219: 4218: 4196: 4193: 4192: 4167: 4164: 4163: 4138: 4135: 4134: 4131: 4093: 4089: 4074: 4070: 4040: 4036: 4021: 4017: 4015: 4012: 4011: 3995: 3991: 3982: 3978: 3976: 3973: 3972: 3936: 3935: 3920: 3916: 3907: 3903: 3900: 3899: 3893: 3892: 3883: 3879: 3870: 3866: 3863: 3862: 3853: 3849: 3840: 3836: 3829: 3828: 3811: 3808: 3807: 3790: 3786: 3784: 3781: 3780: 3773: 3751: 3747: 3745: 3742: 3741: 3718: 3714: 3699: 3695: 3693: 3690: 3689: 3663: 3659: 3650: 3646: 3628: 3624: 3615: 3611: 3609: 3606: 3605: 3589: 3586: 3585: 3561: 3560: 3545: 3541: 3532: 3528: 3525: 3524: 3518: 3517: 3508: 3504: 3495: 3491: 3488: 3487: 3478: 3474: 3465: 3461: 3454: 3453: 3436: 3433: 3432: 3407: 3403: 3388: 3384: 3382: 3379: 3378: 3358: 3355: 3354: 3337: 3333: 3331: 3328: 3327: 3275: 3269: 3265: 3253: 3249: 3240: 3236: 3234: 3231: 3230: 3214: 3211: 3210: 3193: 3189: 3187: 3184: 3183: 3166: 3162: 3160: 3157: 3156: 3149: 3128: 3125: 3124: 3100: 3097: 3096: 3072: 3071: 3059: 3055: 3052: 3051: 3045: 3044: 3038: 3034: 3031: 3030: 3024: 3020: 3013: 3012: 3005: 3004: 2992: 2981: 2975: 2970: 2964: 2953: 2947: 2935: 2931: 2929: 2923: 2922: 2917: 2912: 2907: 2902: 2896: 2895: 2883: 2878: 2872: 2867: 2861: 2856: 2850: 2844: 2840: 2838: 2832: 2831: 2819: 2814: 2808: 2803: 2797: 2792: 2786: 2780: 2776: 2774: 2764: 2763: 2737: 2734: 2733: 2717: 2714: 2713: 2697: 2694: 2693: 2671: 2668: 2667: 2633: 2630: 2629: 2609: 2606: 2605: 2587: 2586: 2571: 2567: 2558: 2554: 2551: 2550: 2544: 2543: 2534: 2530: 2521: 2517: 2514: 2513: 2504: 2500: 2491: 2487: 2480: 2479: 2462: 2459: 2458: 2441: 2437: 2428: 2424: 2416: 2413: 2412: 2396: 2393: 2392: 2369: 2365: 2350: 2346: 2344: 2341: 2340: 2324: 2321: 2320: 2303: 2299: 2297: 2294: 2293: 2277: 2274: 2273: 2252: 2248: 2242: 2238: 2226: 2215: 2193: 2189: 2187: 2184: 2183: 2166: 2162: 2160: 2157: 2156: 2139: 2135: 2117: 2113: 2098: 2094: 2083: 2080: 2079: 2072: 2064:systematic code 2038: 2034: 2019: 2015: 2013: 2010: 2009: 1968: 1964: 1962: 1959: 1958: 1928: 1905: 1902: 1901: 1900:code satisfies 1894:Singleton bound 1874: 1863: 1860: 1859: 1828: 1802: 1788: 1768: 1757: 1754: 1753: 1719: 1716: 1715: 1689: 1686: 1685: 1633: 1630: 1629: 1607: 1604: 1603: 1587: 1584: 1583: 1558: 1557: 1545: 1537: 1527: 1526: 1519: 1518: 1509: 1505: 1481: 1477: 1459: 1455: 1443: 1442: 1435: 1434: 1426: 1424: 1421: 1420: 1404: 1402: 1399: 1398: 1382:}, ... , where 1326: 1322: 1307: 1303: 1301: 1298: 1297: 1269: 1266: 1265: 1249: 1246: 1245: 1225: 1222: 1221: 1205: 1202: 1201: 1194: 1161: 1158: 1157: 1135: 1132: 1131: 1110: 1102: 1099: 1098: 1075: 1072: 1071: 1055: 1052: 1051: 1035: 1032: 1031: 1000: 997: 996: 977: 974: 973: 954: 951: 950: 931: 928: 927: 920: 814:Pioneer, Venus 745:Mars Pathfinder 721:Voyager program 705: 627: 592: 536: 531: 509:was developed. 483:(DVB) standard 469:digital storage 455:in the form of 453:Voyager program 424:Elwyn Berlekamp 376:Gustave Solomon 368: 352: 348: 341: 333: 321: 313: 309: 305: 301: 239:Gustave Solomon 229:are a group of 220: 176: 172: 158: 135: 67: 65:Polynomial code 63: 46:Gustave Solomon 28: 23: 22: 15: 12: 11: 5: 22582: 22572: 22571: 22566: 22552: 22551: 22546: 22541: 22536: 22531: 22526: 22521: 22514: 22511: 22510: 22509: 22496: 22486: 22471: 22466: 22461: 22456: 22451: 22443: 22440: 22438: 22437:External links 22435: 22434: 22433: 22428: 22410: 22405: 22386: 22371: 22362:(4): 549–557, 22346: 22326: 22321: 22305: 22285: 22271: 22262:(2): 300–304, 22244: 22235:(4): 459–470, 22221: 22185: 22180: 22167: 22158:(8): 2324–33, 22140: 22110: 22107: 22104: 22103: 22067: 22053:10.1.1.13.2021 22028: 21996: 21989: 21949: 21942: 21902: 21889:10.1.1.115.292 21864: 21855: 21843: 21834: 21813:(8): 873–883, 21790: 21765: 21752: 21707: 21692: 21674: 21668: 21643: 21620: 21587: 21572: 21552: 21529: 21510:(2): 207–214. 21494: 21482: 21481: 21479: 21476: 21473: 21472: 21468:bit error rate 21458: 21457: 21455: 21452: 21451: 21450: 21445: 21440: 21435: 21430: 21425: 21420: 21413: 21410: 21402: 21335: 21315: 21287: 21246: 21233: 21211: 21198: 21195: 21194: 21191: 21188: 21184: 21183: 21180: 21177: 21173: 21172: 21169: 21166: 21162: 21161: 21158: 21155: 21151: 21150: 21145: 21142: 21137: 21134: 21126: 21125: 21114: 21111: 21106: 21102: 21091: 21080: 21077: 21072: 21069: 21065: 21054: 21035: 21032: 21027: 21023: 21019: 21016: 21013: 21008: 21004: 21000: 20980: 20975: 20971: 20960: 20949: 20944: 20940: 20936: 20933: 20930: 20925: 20920: 20917: 20914: 20910: 20906: 20901: 20898: 20894: 20878: 20875: 20870: 20867: 20859: 20806: 20786: 20758: 20744: 20738: 20735: 20734: 20731: 20728: 20727: 20724: 20721: 20720: 20717: 20714: 20713: 20710: 20707: 20706: 20703: 20700: 20699: 20696: 20693: 20692: 20690: 20685: 20680: 20672: 20668: 20664: 20663: 20658: 20654: 20650: 20649: 20644: 20640: 20636: 20635: 20630: 20626: 20622: 20621: 20616: 20612: 20608: 20607: 20602: 20598: 20594: 20593: 20588: 20584: 20580: 20579: 20577: 20570: 20564: 20561: 20559: 20556: 20554: 20551: 20549: 20546: 20544: 20541: 20539: 20536: 20534: 20531: 20530: 20527: 20524: 20522: 20519: 20517: 20514: 20512: 20509: 20507: 20504: 20502: 20499: 20497: 20494: 20493: 20490: 20487: 20485: 20482: 20480: 20477: 20475: 20472: 20470: 20467: 20465: 20462: 20460: 20457: 20456: 20453: 20450: 20448: 20445: 20443: 20440: 20438: 20435: 20433: 20430: 20428: 20425: 20423: 20420: 20419: 20416: 20413: 20411: 20408: 20406: 20403: 20401: 20398: 20396: 20393: 20391: 20388: 20386: 20383: 20382: 20379: 20376: 20374: 20371: 20369: 20366: 20364: 20361: 20359: 20356: 20354: 20351: 20349: 20346: 20345: 20342: 20339: 20337: 20334: 20332: 20329: 20327: 20324: 20322: 20319: 20317: 20314: 20312: 20309: 20308: 20306: 20274: 20268: 20265: 20264: 20261: 20258: 20257: 20254: 20251: 20250: 20247: 20244: 20243: 20240: 20237: 20236: 20233: 20230: 20229: 20226: 20223: 20222: 20220: 20215: 20210: 20202: 20198: 20194: 20193: 20188: 20184: 20180: 20179: 20174: 20170: 20166: 20165: 20160: 20156: 20152: 20151: 20146: 20142: 20138: 20137: 20132: 20128: 20124: 20123: 20118: 20114: 20110: 20109: 20107: 20100: 20094: 20091: 20089: 20086: 20084: 20081: 20079: 20076: 20074: 20071: 20069: 20066: 20064: 20061: 20060: 20057: 20054: 20052: 20049: 20047: 20044: 20042: 20039: 20037: 20034: 20032: 20029: 20027: 20024: 20023: 20020: 20017: 20015: 20012: 20010: 20007: 20005: 20002: 20000: 19997: 19995: 19992: 19990: 19987: 19986: 19983: 19980: 19978: 19975: 19973: 19970: 19968: 19965: 19963: 19960: 19958: 19955: 19953: 19950: 19949: 19946: 19943: 19941: 19938: 19936: 19933: 19931: 19928: 19926: 19923: 19921: 19918: 19916: 19913: 19912: 19909: 19906: 19904: 19901: 19899: 19896: 19894: 19891: 19889: 19886: 19884: 19881: 19879: 19876: 19875: 19872: 19869: 19867: 19864: 19862: 19859: 19857: 19854: 19852: 19849: 19847: 19844: 19842: 19839: 19838: 19836: 19811: 19806: 19802: 19796: 19792: 19788: 19785: 19782: 19777: 19772: 19768: 19762: 19758: 19754: 19749: 19744: 19740: 19734: 19730: 19726: 19721: 19716: 19712: 19706: 19702: 19698: 19693: 19689: 19683: 19679: 19675: 19670: 19666: 19662: 19659: 19656: 19651: 19647: 19641: 19637: 19633: 19628: 19624: 19620: 19615: 19611: 19582: 19579: 19576: 19571: 19567: 19563: 19560: 19557: 19554: 19549: 19545: 19541: 19538: 19533: 19529: 19500: 19489: 19466: 19455: 19444: 19437: 19434: 19421: 19401: 19396: 19392: 19388: 19385: 19368: 19365: 19352: 19349: 19346: 19343: 19340: 19317: 19314: 19311: 19308: 19305: 19302: 19299: 19296: 19291: 19288: 19282: 19276: 19271: 19268: 19263: 19233: 19229: 19206: 19202: 19186: 19183: 19177: 19174: 17969:% polynomials) 17648: 17642: 17639: 17347: 17337: 17334: 17332: 17331:MATLAB example 17329: 17298: 17295: 17265: 17260: 17256: 17252: 17249: 17246: 17203:−1) / 2⌋ 17163: 17160: 17101: 17098: 17095: 17092: 17089: 17081: 17079: 17074: 17071: 17068: 17064: 17058: 17054: 17050: 17047: 17044: 17039: 17036: 17033: 17029: 17023: 17019: 17015: 17010: 17007: 17004: 17000: 16994: 16990: 16986: 16983: 16980: 16977: 16975: 16971: 16967: 16963: 16962: 16959: 16954: 16950: 16944: 16941: 16938: 16934: 16930: 16927: 16924: 16919: 16916: 16913: 16909: 16903: 16899: 16895: 16890: 16886: 16882: 16875: 16871: 16867: 16862: 16859: 16856: 16854: 16850: 16846: 16842: 16841: 16819: 16815: 16792: 16788: 16750: 16746: 16723: 16719: 16696: 16693: 16690: 16687: 16684: 16675: 16670: 16666: 16662: 16659: 16656: 16651: 16647: 16643: 16638: 16634: 16630: 16625: 16621: 16441: 16438: 16429: 16420: 16407: 16398: 16395: 16394: 16383: 16376: 16372: 16371: 16364: 16353: 16349: 16348: 16345: 16330: 16326: 16325: 16322: 16303: 16299: 16298: 16293: 16290: 16285: 16282: 16270: 16267: 16262: 16256: 16245: 16236: 16227: 16220: 16211: 16202: 16195: 16171: 16158: 16149: 16140: 16127: 16118: 16109: 16099: 16074: 16059: 16052: 16037: 16027: 16023: 15990: 15986: 15971: 15967: 15952: 15948: 15933: 15909: 15892: 15871: 15867: 15847: 15839: 15835: 15831: 15830: 15827: 15822: 15818: 15814: 15813: 15808: 15804: 15798: 15794: 15790: 15789: 15786: 15783: 15782: 15779: 15776: 15775: 15772: 15769: 15768: 15763: 15759: 15753: 15749: 15745: 15744: 15739: 15735: 15729: 15725: 15721: 15720: 15715: 15711: 15705: 15701: 15697: 15696: 15694: 15689: 15684: 15676: 15672: 15668: 15667: 15664: 15661: 15657: 15653: 15647: 15643: 15639: 15634: 15630: 15626: 15625: 15620: 15616: 15612: 15608: 15604: 15598: 15594: 15590: 15585: 15581: 15575: 15571: 15567: 15562: 15558: 15554: 15553: 15548: 15544: 15540: 15536: 15532: 15526: 15522: 15518: 15513: 15509: 15503: 15499: 15495: 15490: 15486: 15480: 15476: 15472: 15467: 15463: 15459: 15458: 15453: 15449: 15445: 15441: 15437: 15431: 15427: 15423: 15418: 15414: 15408: 15404: 15400: 15395: 15391: 15385: 15381: 15377: 15372: 15368: 15364: 15363: 15358: 15354: 15350: 15346: 15342: 15336: 15332: 15328: 15323: 15319: 15313: 15309: 15305: 15300: 15296: 15290: 15286: 15282: 15277: 15273: 15269: 15268: 15263: 15259: 15255: 15251: 15247: 15241: 15237: 15233: 15228: 15224: 15218: 15214: 15210: 15205: 15201: 15195: 15191: 15187: 15186: 15181: 15177: 15173: 15169: 15165: 15159: 15155: 15151: 15146: 15142: 15136: 15132: 15128: 15127: 15122: 15118: 15114: 15110: 15106: 15100: 15096: 15092: 15091: 15089: 15057: 15054: 15051: 15048: 15045: 15040: 15036: 15032: 15029: 15026: 15023: 15020: 15017: 15014: 15011: 15008: 15005: 15002: 14999: 14996: 14968: 14964: 14960: 14957: 14952: 14948: 14944: 14941: 14938: 14933: 14930: 14927: 14923: 14917: 14914: 14911: 14907: 14903: 14898: 14894: 14888: 14884: 14880: 14877: 14875: 14873: 14870: 14867: 14864: 14861: 14860: 14857: 14854: 14851: 14846: 14842: 14838: 14835: 14832: 14827: 14824: 14821: 14817: 14811: 14808: 14805: 14801: 14797: 14792: 14788: 14782: 14778: 14774: 14771: 14769: 14767: 14764: 14761: 14758: 14755: 14754: 14749: 14745: 14741: 14738: 14733: 14729: 14725: 14722: 14719: 14714: 14711: 14708: 14704: 14698: 14695: 14692: 14688: 14684: 14679: 14676: 14673: 14669: 14663: 14659: 14655: 14652: 14650: 14648: 14645: 14642: 14639: 14636: 14635: 14622:syndromes and 14592: 14589: 14575: 14574: 14571: 14568: 14561: 14550: 14547: 14544: 14540: 14539: 14536: 14533: 14526: 14515: 14512: 14509: 14505: 14504: 14501: 14498: 14495: 14488: 14485: 14482: 14478: 14477: 14474: 14471: 14468: 14461: 14458: 14455: 14451: 14450: 14445: 14440: 14435: 14430: 14425: 14419: 14414: 14402: 14399: 14360: 14357: 14354: 14350: 14341: 14337: 14333: 14330: 14327: 14322: 14319: 14316: 14312: 14303: 14299: 14295: 14290: 14286: 14282: 14279: 14255: 14252: 14225: 14221: 14217: 14214: 14209: 14205: 14201: 14198: 14193: 14189: 14183: 14179: 14175: 14170: 14166: 14160: 14156: 14140: 14136: 14132: 14130: 14125: 14121: 14117: 14115: 14112: 14109: 14096: 14088: 14084:= 757 = 3 and 14081: 14076: 14062: 14056: 14053: 14052: 14049: 14046: 14045: 14043: 14038: 14033: 14025: 14021: 14017: 14016: 14011: 14007: 14003: 14002: 14000: 13993: 13987: 13984: 13982: 13979: 13978: 13975: 13972: 13970: 13967: 13966: 13964: 13934: 13928: 13925: 13924: 13921: 13918: 13917: 13915: 13910: 13905: 13899: 13896: 13893: 13892: 13889: 13886: 13883: 13882: 13880: 13875: 13870: 13862: 13858: 13854: 13853: 13848: 13844: 13840: 13839: 13837: 13830: 13824: 13821: 13819: 13816: 13815: 13812: 13809: 13807: 13804: 13803: 13801: 13780: 13777: 13774: 13769: 13765: 13761: 13758: 13755: 13750: 13746: 13741: 13738: 13735: 13732: 13727: 13723: 13719: 13716: 13713: 13708: 13704: 13699: 13696: 13693: 13690: 13685: 13681: 13677: 13674: 13671: 13666: 13662: 13642: 13639: 13636: 13633: 13630: 13627: 13624: 13621: 13616: 13612: 13608: 13605: 13602: 13597: 13593: 13589: 13586: 13583: 13578: 13574: 13570: 13567: 13564: 13559: 13555: 13551: 13548: 13545: 13540: 13536: 13532: 13529: 13526: 13523: 13518: 13514: 13510: 13507: 13504: 13499: 13495: 13466: 13463: 13460: 13457: 13454: 13449: 13445: 13441: 13438: 13433: 13429: 13425: 13422: 13417: 13413: 13409: 13406: 13401: 13397: 13393: 13390: 13385: 13381: 13377: 13374: 13371: 13368: 13365: 13362: 13359: 13356: 13353: 13350: 13347: 13344: 13341: 13338: 13335: 13332: 13312: 13309: 13306: 13303: 13300: 13295: 13291: 13287: 13284: 13279: 13275: 13271: 13268: 13263: 13259: 13255: 13252: 13247: 13243: 13239: 13236: 13231: 13227: 13223: 13220: 13217: 13214: 13211: 13206: 13202: 13198: 13193: 13189: 13184: 13181: 13178: 13175: 13172: 13169: 13166: 13163: 13160: 13141: 13138: 13135: 13132: 13129: 13124: 13120: 13116: 13113: 13108: 13104: 13100: 13097: 13094: 13091: 13088: 13083: 13079: 13073: 13069: 13064: 13061: 13058: 13055: 13052: 13049: 13046: 13043: 13038: 13034: 12994: 12991: 12988: 12985: 12982: 12977: 12973: 12969: 12966: 12961: 12957: 12953: 12950: 12945: 12941: 12937: 12934: 12929: 12925: 12921: 12918: 12915: 12912: 12907: 12903: 12899: 12896: 12893: 12890: 12885: 12881: 12877: 12874: 12871: 12868: 12865: 12862: 12859: 12856: 12853: 12850: 12847: 12844: 12841: 12804: 12801: 12778: 12776: 12769: 12755: 12754:Fix the errors 12752: 12746: 12732: 12719: 12713: 12710: 12696: 12689: 12683: 12680: 12661: 12658: 12653: 12649: 12626: 12622: 12608: 12600: 12596: 12593: 12591:, p. 35) 12557: 12549: 12546: 12543: 12539: 12535: 12532: 12531: 12528: 12525: 12524: 12519: 12516: 12513: 12509: 12505: 12502: 12501: 12496: 12493: 12490: 12486: 12482: 12479: 12478: 12476: 12471: 12466: 12458: 12454: 12450: 12449: 12446: 12443: 12442: 12437: 12434: 12431: 12427: 12423: 12422: 12417: 12413: 12409: 12408: 12406: 12399: 12391: 12388: 12385: 12382: 12378: 12374: 12372: 12369: 12365: 12362: 12359: 12355: 12351: 12347: 12343: 12339: 12338: 12335: 12332: 12330: 12327: 12325: 12322: 12320: 12317: 12316: 12311: 12308: 12305: 12301: 12297: 12295: 12292: 12288: 12284: 12280: 12276: 12272: 12268: 12267: 12262: 12258: 12254: 12252: 12249: 12245: 12241: 12237: 12233: 12229: 12225: 12224: 12222: 12206: 12178: 12175: 12172: 12168: 12164: 12161: 12156: 12152: 12146: 12143: 12140: 12137: 12134: 12130: 12126: 12123: 12120: 12115: 12112: 12109: 12105: 12099: 12096: 12093: 12089: 12085: 12080: 12076: 12070: 12066: 12043: 12040: 12037: 12033: 12010: 12007: 12002: 11998: 11992: 11988: 11984: 11979: 11976: 11973: 11969: 11963: 11960: 11957: 11953: 11949: 11946: 11943: 11938: 11935: 11932: 11929: 11926: 11922: 11916: 11912: 11908: 11903: 11900: 11897: 11893: 11870: 11867: 11863: 11857: 11852: 11848: 11842: 11838: 11832: 11827: 11824: 11821: 11817: 11812: 11806: 11802: 11798: 11795: 11792: 11788: 11782: 11779: 11776: 11773: 11770: 11765: 11761: 11755: 11751: 11745: 11740: 11737: 11734: 11730: 11725: 11719: 11715: 11711: 11707: 11701: 11698: 11695: 11692: 11689: 11684: 11680: 11674: 11670: 11664: 11659: 11656: 11653: 11649: 11644: 11638: 11634: 11630: 11626: 11620: 11617: 11614: 11609: 11605: 11599: 11595: 11589: 11584: 11581: 11578: 11574: 11569: 11548: 11526: 11523: 11519: 11513: 11508: 11504: 11498: 11494: 11488: 11484: 11478: 11473: 11470: 11467: 11463: 11458: 11454: 11451: 11448: 11444: 11438: 11435: 11432: 11429: 11426: 11421: 11417: 11411: 11407: 11401: 11397: 11391: 11386: 11383: 11380: 11376: 11371: 11367: 11363: 11357: 11354: 11351: 11348: 11345: 11340: 11336: 11330: 11326: 11320: 11316: 11310: 11305: 11302: 11299: 11295: 11290: 11286: 11282: 11276: 11273: 11270: 11265: 11261: 11255: 11251: 11245: 11240: 11237: 11234: 11230: 11225: 11202: 11199: 11195: 11189: 11184: 11180: 11174: 11170: 11164: 11160: 11156: 11153: 11150: 11145: 11142: 11139: 11136: 11133: 11128: 11124: 11118: 11114: 11108: 11104: 11100: 11095: 11092: 11089: 11086: 11083: 11078: 11074: 11068: 11064: 11058: 11054: 11050: 11045: 11042: 11039: 11034: 11030: 11024: 11020: 11015: 11009: 11004: 11001: 10998: 10994: 10959: 10956: 10951: 10946: 10942: 10936: 10932: 10926: 10922: 10918: 10915: 10912: 10907: 10904: 10901: 10898: 10895: 10890: 10886: 10880: 10876: 10870: 10866: 10862: 10857: 10854: 10851: 10848: 10845: 10840: 10836: 10830: 10826: 10820: 10816: 10812: 10807: 10804: 10801: 10796: 10792: 10786: 10782: 10778: 10776: 10773: 10770: 10765: 10762: 10757: 10753: 10747: 10744: 10741: 10736: 10732: 10726: 10722: 10716: 10712: 10708: 10705: 10702: 10697: 10694: 10689: 10685: 10679: 10676: 10673: 10668: 10664: 10658: 10654: 10648: 10644: 10640: 10635: 10632: 10627: 10623: 10617: 10614: 10611: 10606: 10602: 10596: 10592: 10586: 10582: 10578: 10573: 10570: 10567: 10562: 10558: 10552: 10548: 10544: 10542: 10539: 10536: 10532: 10526: 10523: 10518: 10514: 10508: 10504: 10500: 10497: 10494: 10489: 10486: 10481: 10477: 10471: 10467: 10463: 10458: 10455: 10450: 10446: 10440: 10436: 10432: 10429: 10425: 10419: 10416: 10413: 10408: 10404: 10398: 10394: 10390: 10388: 10385: 10382: 10379: 10374: 10371: 10366: 10362: 10358: 10355: 10350: 10347: 10344: 10339: 10335: 10329: 10325: 10321: 10319: 10297: 10294: 10291: 10286: 10282: 10276: 10272: 10251: 10248: 10245: 10242: 10239: 10219: 10197: 10194: 10191: 10186: 10183: 10178: 10174: 10170: 10167: 10147: 10144: 10141: 10138: 10135: 10132: 10129: 10124: 10120: 10116: 10111: 10108: 10103: 10099: 10095: 10092: 10089: 10067: 10064: 10059: 10055: 10051: 10048: 10026: 10023: 10018: 10014: 9981: 9977: 9971: 9967: 9963: 9960: 9957: 9952: 9948: 9942: 9938: 9934: 9929: 9925: 9919: 9915: 9911: 9908: 9905: 9902: 9897: 9893: 9889: 9886: 9883: 9880: 9875: 9870: 9867: 9864: 9860: 9856: 9853: 9850: 9847: 9844: 9817: 9811: 9808: 9795: 9775: 9772: 9749: 9746: 9743: 9730: 9717: 9711: 9693: 9685: 9682: 9679: 9675: 9671: 9670: 9667: 9664: 9663: 9658: 9654: 9650: 9649: 9644: 9640: 9636: 9635: 9633: 9628: 9623: 9615: 9611: 9607: 9606: 9603: 9600: 9599: 9594: 9590: 9586: 9585: 9580: 9576: 9572: 9571: 9569: 9562: 9554: 9551: 9548: 9543: 9539: 9535: 9533: 9530: 9526: 9523: 9520: 9515: 9511: 9507: 9503: 9500: 9497: 9492: 9488: 9484: 9483: 9480: 9477: 9475: 9472: 9470: 9467: 9465: 9462: 9461: 9456: 9451: 9447: 9443: 9441: 9438: 9434: 9429: 9425: 9421: 9417: 9412: 9408: 9404: 9403: 9398: 9393: 9389: 9385: 9383: 9380: 9376: 9371: 9367: 9363: 9359: 9354: 9350: 9346: 9345: 9343: 9332:error values. 9328: 9321: 9314: 9307:equations in 2 9277: 9272: 9268: 9264: 9259: 9255: 9247: 9243: 9238: 9234: 9231: 9224: 9220: 9216: 9213: 9209: 9205: 9198: 9194: 9189: 9183: 9179: 9175: 9151: 9146: 9142: 9136: 9132: 9126: 9121: 9118: 9115: 9111: 9107: 9102: 9098: 9071: 9067: 9062: 9058: 9053: 9049: 9042: 9035: 9031: 9026: 9022: 9017: 9013: 8999: 8989: 8985:error locators 8980: 8977: 8958: 8954: 8933: 8930: 8927: 8924: 8904: 8901: 8898: 8893: 8889: 8885: 8882: 8858: 8855: 8852: 8849: 8846: 8843: 8840: 8837: 8834: 8831: 8828: 8824: 8817: 8813: 8807: 8802: 8798: 8794: 8785: 8781: 8776: 8770: 8765: 8762: 8759: 8755: 8751: 8748: 8746: 8744: 8741: 8736: 8732: 8728: 8725: 8722: 8719: 8717: 8715: 8712: 8707: 8703: 8699: 8696: 8693: 8690: 8687: 8684: 8679: 8675: 8671: 8668: 8665: 8662: 8657: 8653: 8649: 8646: 8643: 8640: 8635: 8631: 8627: 8624: 8621: 8618: 8616: 8612: 8608: 8604: 8603: 8589: 8572: 8569: 8566: 8562: 8558: 8553: 8549: 8536: 8533: 8502: 8500: 8493: 8469: 8465: 8460: 8452: 8448: 8443: 8437: 8432: 8429: 8426: 8422: 8418: 8415: 8412: 8409: 8406: 8389: 8374: 8356: 8352: 8346: 8342: 8336: 8333: 8330: 8325: 8322: 8319: 8315: 8311: 8308: 8305: 8302: 8299: 8280: 8277: 8274: 8271: 8268: 8265: 8262: 8259: 8256: 8253: 8250: 8247: 8244: 8241: 8203: 8200: 8197: 8194: 8191: 8188: 8185: 8182: 8179: 8176: 8173: 8167: 8164: 8161: 8158: 8153: 8149: 8145: 8142: 8122: 8119: 8116: 8111: 8107: 8103: 8100: 8095: 8091: 8087: 8084: 8081: 8078: 8075: 8072: 8067: 8063: 8059: 8056: 8051: 8047: 8043: 8040: 8037: 8034: 7991: 7988: 7983: 7979: 7975: 7972: 7969: 7964: 7961: 7958: 7953: 7950: 7947: 7943: 7939: 7936: 7933: 7930: 7927: 7887: 7883: 7877: 7873: 7867: 7864: 7861: 7856: 7853: 7850: 7846: 7842: 7839: 7836: 7833: 7830: 7802: 7797: 7794: 7791: 7787: 7783: 7780: 7777: 7772: 7768: 7764: 7761: 7758: 7753: 7749: 7745: 7733: 7730: 7713:Main article: 7710: 7707: 7702: 7699: 7682: 7679: 7658: 7632: 7628: 7624: 7619: 7615: 7594: 7591: 7586: 7582: 7578: 7575: 7572: 7569: 7566: 7563: 7558: 7554: 7550: 7547: 7544: 7524: 7521: 7518: 7515: 7512: 7509: 7506: 7486: 7483: 7478: 7475: 7472: 7468: 7464: 7461: 7458: 7455: 7452: 7449: 7444: 7440: 7436: 7433: 7430: 7410: 7390: 7369: 7365: 7361: 7357: 7354: 7334: 7331: 7328: 7325: 7322: 7319: 7316: 7313: 7310: 7307: 7304: 7282: 7278: 7268:take the form 7257: 7237: 7226:primitive root 7213: 7163: 7160: 7157: 7154: 7151: 7148: 7145: 7142: 7139: 7136: 7133: 7113: 7110: 7107: 7087: 7084: 7081: 7057: 7054: 7051: 7031: 7007: 7004: 7001: 6998: 6993: 6989: 6985: 6982: 6962: 6959: 6954: 6950: 6946: 6943: 6923: 6901: 6897: 6876: 6853: 6833: 6810: 6807: 6802: 6798: 6793: 6788: 6785: 6763: 6759: 6755: 6752: 6749: 6746: 6743: 6740: 6737: 6732: 6728: 6707: 6704: 6701: 6696: 6692: 6688: 6683: 6680: 6675: 6672: 6650: 6647: 6644: 6640: 6634: 6630: 6626: 6623: 6620: 6615: 6610: 6606: 6599: 6594: 6591: 6586: 6580: 6575: 6570: 6567: 6564: 6561: 6558: 6554: 6548: 6545: 6538: 6535: 6530: 6525: 6521: 6498: 6495: 6492: 6488: 6482: 6478: 6474: 6471: 6468: 6463: 6458: 6454: 6447: 6442: 6439: 6434: 6428: 6423: 6418: 6415: 6412: 6409: 6406: 6402: 6396: 6393: 6385: 6382: 6377: 6373: 6366: 6363: 6360: 6356: 6350: 6345: 6341: 6301: 6281: 6227: 6223: 6219: 6216: 6213: 6210: 6207: 6187: 6184: 6181: 6142: 6139: 6136: 6133: 6130: 6127: 6122: 6118: 6083: 6080: 6068: 6064: 6061: 6058: 6055: 6050: 6046: 6043: 6040: 6037: 6034: 6029: 6025: 6021: 6018: 6015: 6012: 6007: 6003: 5999: 5996: 5993: 5990: 5985: 5981: 5977: 5972: 5968: 5964: 5961: 5958: 5955: 5952: 5949: 5946: 5943: 5940: 5937: 5917: 5914: 5911: 5908: 5887: 5866: 5863: 5860: 5857: 5835: 5831: 5828: 5825: 5820: 5816: 5812: 5807: 5803: 5799: 5796: 5793: 5790: 5787: 5784: 5781: 5778: 5775: 5772: 5752: 5749: 5746: 5743: 5723: 5703: 5700: 5697: 5694: 5672: 5668: 5664: 5661: 5658: 5655: 5652: 5630: 5626: 5622: 5617: 5613: 5609: 5606: 5603: 5598: 5595: 5592: 5588: 5584: 5579: 5576: 5573: 5569: 5548: 5545: 5542: 5520: 5517: 5514: 5511: 5508: 5499: 5490: 5486: 5482: 5479: 5476: 5473: 5470: 5467: 5464: 5461: 5458: 5453: 5449: 5428: 5425: 5422: 5417: 5413: 5392: 5372: 5369: 5366: 5363: 5343: 5340: 5337: 5334: 5331: 5309: 5305: 5284: 5281: 5278: 5275: 5252: 5249: 5246: 5243: 5223: 5220: 5217: 5214: 5194: 5191: 5188: 5185: 5165: 5162: 5159: 5156: 5136: 5133: 5130: 5127: 5107: 5104: 5101: 5098: 5078: 5075: 5072: 5069: 5049: 5029: 5026: 5023: 5020: 5000: 4997: 4994: 4991: 4988: 4985: 4982: 4979: 4976: 4973: 4970: 4967: 4964: 4948: 4945: 4933: 4929: 4923: 4920: 4917: 4913: 4909: 4906: 4903: 4898: 4894: 4890: 4885: 4881: 4870: 4866: 4860: 4856: 4850: 4845: 4842: 4839: 4835: 4831: 4828: 4825: 4822: 4819: 4813: 4806: 4800: 4796: 4792: 4789: 4786: 4781: 4777: 4773: 4768: 4764: 4759: 4754: 4750: 4746: 4723: 4720: 4717: 4693: 4690: 4687: 4683: 4679: 4674: 4671: 4668: 4665: 4662: 4658: 4652: 4649: 4646: 4643: 4640: 4636: 4632: 4629: 4626: 4623: 4618: 4614: 4610: 4605: 4601: 4597: 4593: 4587: 4584: 4581: 4578: 4575: 4572: 4569: 4565: 4561: 4558: 4554: 4550: 4546: 4540: 4537: 4534: 4530: 4526: 4523: 4519: 4514: 4508: 4504: 4500: 4497: 4493: 4489: 4486: 4483: 4480: 4477: 4458: 4438: 4435: 4432: 4429: 4409: 4406: 4403: 4383: 4380: 4377: 4374: 4351: 4348: 4345: 4325: 4322: 4319: 4316: 4296: 4293: 4290: 4287: 4267: 4264: 4261: 4258: 4238: 4235: 4232: 4229: 4226: 4206: 4203: 4200: 4180: 4177: 4174: 4171: 4151: 4148: 4145: 4142: 4130: 4127: 4107: 4102: 4099: 4096: 4092: 4088: 4085: 4082: 4077: 4073: 4069: 4066: 4063: 4060: 4057: 4054: 4049: 4046: 4043: 4039: 4035: 4032: 4029: 4024: 4020: 3998: 3994: 3990: 3985: 3981: 3940: 3934: 3929: 3926: 3923: 3919: 3915: 3910: 3906: 3902: 3901: 3898: 3895: 3894: 3891: 3886: 3882: 3878: 3873: 3869: 3865: 3864: 3861: 3856: 3852: 3848: 3843: 3839: 3835: 3834: 3832: 3827: 3824: 3821: 3818: 3815: 3793: 3789: 3772: 3769: 3754: 3750: 3727: 3724: 3721: 3717: 3713: 3710: 3707: 3702: 3698: 3688:, are exactly 3677: 3672: 3669: 3666: 3662: 3658: 3653: 3649: 3645: 3642: 3639: 3636: 3631: 3627: 3623: 3618: 3614: 3593: 3565: 3559: 3554: 3551: 3548: 3544: 3540: 3535: 3531: 3527: 3526: 3523: 3520: 3519: 3516: 3511: 3507: 3503: 3498: 3494: 3490: 3489: 3486: 3481: 3477: 3473: 3468: 3464: 3460: 3459: 3457: 3452: 3449: 3446: 3443: 3440: 3416: 3413: 3410: 3406: 3402: 3399: 3396: 3391: 3387: 3373:, one can use 3362: 3340: 3336: 3315: 3312: 3309: 3306: 3303: 3300: 3297: 3294: 3291: 3288: 3285: 3282: 3272: 3268: 3264: 3261: 3256: 3252: 3248: 3243: 3239: 3218: 3196: 3192: 3169: 3165: 3148: 3145: 3132: 3104: 3076: 3068: 3065: 3062: 3058: 3054: 3053: 3050: 3047: 3046: 3041: 3037: 3033: 3032: 3027: 3023: 3019: 3018: 3016: 3009: 3001: 2998: 2995: 2990: 2987: 2984: 2980: 2976: 2974: 2971: 2967: 2962: 2959: 2956: 2952: 2948: 2944: 2941: 2938: 2934: 2930: 2928: 2925: 2924: 2921: 2918: 2916: 2913: 2911: 2908: 2906: 2903: 2901: 2898: 2897: 2892: 2889: 2886: 2881: 2877: 2873: 2871: 2868: 2864: 2859: 2855: 2851: 2847: 2843: 2839: 2837: 2834: 2833: 2828: 2825: 2822: 2817: 2813: 2809: 2807: 2804: 2800: 2795: 2791: 2787: 2783: 2779: 2775: 2773: 2770: 2769: 2767: 2762: 2759: 2756: 2753: 2750: 2747: 2744: 2741: 2721: 2701: 2681: 2678: 2675: 2655: 2652: 2649: 2646: 2643: 2640: 2637: 2626:linear mapping 2613: 2604:This function 2591: 2585: 2580: 2577: 2574: 2570: 2566: 2561: 2557: 2553: 2552: 2549: 2546: 2545: 2542: 2537: 2533: 2529: 2524: 2520: 2516: 2515: 2512: 2507: 2503: 2499: 2494: 2490: 2486: 2485: 2483: 2478: 2475: 2472: 2469: 2466: 2444: 2440: 2436: 2431: 2427: 2423: 2420: 2400: 2378: 2375: 2372: 2368: 2364: 2361: 2358: 2353: 2349: 2328: 2306: 2302: 2281: 2261: 2255: 2251: 2245: 2241: 2235: 2232: 2229: 2224: 2221: 2218: 2214: 2210: 2207: 2204: 2201: 2196: 2192: 2169: 2165: 2142: 2138: 2134: 2131: 2126: 2123: 2120: 2116: 2112: 2109: 2106: 2101: 2097: 2093: 2090: 2087: 2078:, the message 2071: 2068: 2041: 2037: 2033: 2030: 2027: 2022: 2018: 1971: 1967: 1935: 1931: 1927: 1924: 1921: 1918: 1915: 1912: 1909: 1881: 1877: 1873: 1870: 1867: 1847: 1844: 1841: 1838: 1835: 1831: 1827: 1824: 1821: 1818: 1815: 1812: 1809: 1805: 1801: 1798: 1795: 1791: 1787: 1784: 1781: 1778: 1775: 1771: 1767: 1764: 1761: 1741: 1738: 1735: 1732: 1729: 1726: 1723: 1699: 1696: 1693: 1673: 1670: 1667: 1664: 1661: 1658: 1655: 1652: 1649: 1646: 1643: 1640: 1637: 1617: 1614: 1611: 1591: 1578:Since any two 1567: 1561: 1555: 1552: 1544: 1536: 1530: 1522: 1517: 1512: 1508: 1504: 1501: 1498: 1495: 1492: 1489: 1484: 1480: 1476: 1473: 1470: 1467: 1462: 1458: 1454: 1451: 1446: 1438: 1433: 1429: 1407: 1329: 1325: 1321: 1318: 1315: 1310: 1306: 1273: 1253: 1229: 1209: 1193: 1190: 1177: 1174: 1171: 1168: 1165: 1145: 1142: 1139: 1117: 1114: 1109: 1106: 1079: 1059: 1039: 1016: 1013: 1010: 1007: 1004: 981: 971:message length 958: 935: 919: 916: 913: 912: 909: 904: 900: 899: 896: 891: 887: 886: 883: 880: 876: 875: 872: 869: 865: 864: 861: 858: 854: 853: 850: 847: 843: 842: 839: 834: 830: 829: 826: 820: 816: 815: 812: 811:(CC) (25, 1/2) 806: 802: 801: 798: 795: 791: 790: 787: 784: 704: 701: 626: 623: 591: 588: 535: 532: 530: 527: 372:Irving S. Reed 367: 364: 345: + 1 235:Irving S. Reed 222: 221: 219: 218: 211: 204: 196: 193: 192: 186: 185: 181: 180: 166: 165: 161: 160: 154: 152: 146: 145: 117: 111: 110: 100: 94: 93: 88: 86:Message length 82: 81: 76: 70: 69: 58: 54: 53: 52:Classification 49: 48: 42:Irving S. Reed 39: 35: 34: 26: 9: 6: 4: 3: 2: 22581: 22570: 22569:Coding theory 22567: 22565: 22562: 22561: 22559: 22550: 22547: 22545: 22542: 22540: 22537: 22535: 22532: 22530: 22527: 22525: 22522: 22520: 22517: 22516: 22504: 22503: 22497: 22494: 22490: 22487: 22483: 22479: 22478: 22472: 22470: 22467: 22465: 22462: 22460: 22457: 22455: 22452: 22449: 22446: 22445: 22431: 22429:9781461550051 22425: 22421: 22420: 22415: 22411: 22408: 22402: 22398: 22397: 22392: 22387: 22384:on 2013-03-13 22383: 22379: 22378: 22372: 22369: 22365: 22361: 22357: 22356: 22351: 22347: 22343: 22339: 22335: 22331: 22327: 22324: 22318: 22314: 22310: 22306: 22302: 22298: 22294: 22290: 22286: 22279: 22278: 22272: 22269: 22265: 22261: 22257: 22253: 22249: 22245: 22242: 22238: 22234: 22230: 22226: 22222: 22219: 22215: 22211: 22207: 22203: 22202: 22194: 22190: 22189:Massey, J. L. 22186: 22183: 22177: 22173: 22168: 22165: 22161: 22157: 22153: 22146: 22141: 22127: 22120: 22119: 22113: 22112: 22089: 22085: 22078: 22071: 22063: 22059: 22054: 22049: 22045: 22041: 22040: 22032: 22018: 22013: 22009: 22008: 22000: 21992: 21986: 21982: 21978: 21973: 21968: 21964: 21960: 21953: 21945: 21939: 21935: 21931: 21926: 21921: 21917: 21913: 21906: 21899: 21895: 21890: 21885: 21881: 21877: 21876: 21868: 21859: 21852: 21847: 21838: 21831: 21816: 21812: 21808: 21801: 21794: 21779: 21775: 21769: 21762: 21756: 21748: 21744: 21740: 21736: 21732: 21728: 21721: 21714: 21712: 21703: 21699: 21695: 21693:9780470546345 21689: 21685: 21678: 21671: 21665: 21661: 21657: 21653: 21647: 21636: 21635: 21627: 21625: 21615: 21610: 21606: 21602: 21598: 21591: 21583: 21579: 21575: 21569: 21565: 21564: 21556: 21548: 21544: 21541:. MIT Press. 21540: 21533: 21525: 21521: 21517: 21513: 21509: 21505: 21498: 21492: 21487: 21483: 21469: 21463: 21459: 21449: 21446: 21444: 21441: 21439: 21436: 21434: 21431: 21429: 21426: 21424: 21421: 21419: 21416: 21415: 21406: 21401: 21398: 21391: 21387: 21380: 21376: 21367: 21363: 21359: 21355: 21351: 21347: 21343: 21339: 21331: 21327: 21323: 21319: 21311: 21307: 21303: 21299: 21295: 21291: 21286: 21284: 21280: 21276: 21272: 21268: 21264: 21256: 21252: 21245: 21241: 21237: 21229: 21225: 21221: 21217: 21210: 21206: 21202: 21192: 21189: 21186: 21185: 21181: 21178: 21175: 21174: 21170: 21167: 21164: 21163: 21159: 21156: 21153: 21152: 21148: 21143: 21140: 21135: 21133: 21130: 21129: 21112: 21109: 21104: 21100: 21092: 21078: 21075: 21070: 21067: 21063: 21055: 21053: 21049: 21025: 21021: 21014: 21011: 21006: 21002: 20978: 20973: 20969: 20961: 20942: 20938: 20934: 20931: 20923: 20918: 20915: 20912: 20908: 20904: 20899: 20896: 20892: 20884: 20883: 20882: 20874: 20863: 20858: 20855: 20838: 20834: 20830: 20826: 20822: 20818: 20814: 20810: 20802: 20798: 20794: 20790: 20782: 20778: 20774: 20770: 20766: 20762: 20757: 20742: 20736: 20729: 20722: 20715: 20708: 20701: 20694: 20688: 20683: 20678: 20670: 20666: 20656: 20652: 20642: 20638: 20628: 20624: 20614: 20610: 20600: 20596: 20586: 20582: 20575: 20568: 20562: 20557: 20552: 20547: 20542: 20537: 20532: 20525: 20520: 20515: 20510: 20505: 20500: 20495: 20488: 20483: 20478: 20473: 20468: 20463: 20458: 20451: 20446: 20441: 20436: 20431: 20426: 20421: 20414: 20409: 20404: 20399: 20394: 20389: 20384: 20377: 20372: 20367: 20362: 20357: 20352: 20347: 20340: 20335: 20330: 20325: 20320: 20315: 20310: 20304: 20294: 20292: 20287: 20272: 20266: 20259: 20252: 20245: 20238: 20231: 20224: 20218: 20213: 20208: 20200: 20196: 20186: 20182: 20172: 20168: 20158: 20154: 20144: 20140: 20130: 20126: 20116: 20112: 20105: 20098: 20092: 20087: 20082: 20077: 20072: 20067: 20062: 20055: 20050: 20045: 20040: 20035: 20030: 20025: 20018: 20013: 20008: 20003: 19998: 19993: 19988: 19981: 19976: 19971: 19966: 19961: 19956: 19951: 19944: 19939: 19934: 19929: 19924: 19919: 19914: 19907: 19902: 19897: 19892: 19887: 19882: 19877: 19870: 19865: 19860: 19855: 19850: 19845: 19840: 19834: 19824: 19809: 19804: 19800: 19794: 19790: 19786: 19783: 19775: 19770: 19766: 19760: 19756: 19752: 19747: 19742: 19738: 19732: 19728: 19724: 19719: 19714: 19710: 19704: 19700: 19696: 19691: 19687: 19681: 19677: 19673: 19668: 19664: 19657: 19649: 19645: 19639: 19635: 19631: 19626: 19622: 19613: 19609: 19600: 19598: 19593: 19580: 19577: 19569: 19565: 19558: 19555: 19547: 19543: 19536: 19531: 19527: 19518: 19512: 19508: 19504: 19499: 19493: 19488: 19482: 19478: 19474: 19470: 19465: 19459: 19454: 19452: 19447: 19443: 19433: 19419: 19394: 19390: 19383: 19374: 19364: 19347: 19344: 19341: 19315: 19312: 19309: 19303: 19300: 19297: 19289: 19286: 19280: 19269: 19266: 19250: 19231: 19227: 19204: 19200: 19191: 19182: 18668:Syndrome_Vals 18635:Syndrome_Vals 17646: 17345: 17343: 17328: 17325: 17321: 17317: 17316:soft-decision 17313: 17309: 17305: 17297:Soft-decoding 17294: 17291: 17287: 17282: 17281:list decoding 17277: 17258: 17254: 17247: 17244: 17236: 17235:list-decoding 17232: 17227: 17223: 17218: 17214: 17210: 17202: 17196: 17190: 17186: 17181: 17177: 17173: 17169: 17159: 17157: 17153: 17150:) to produce 17149: 17145: 17141: 17137: 17133: 17129: 17125: 17121: 17116: 17099: 17096: 17093: 17090: 17087: 17072: 17069: 17066: 17062: 17056: 17048: 17045: 17042: 17037: 17034: 17031: 17027: 17021: 17013: 17008: 17005: 17002: 16998: 16992: 16981: 16978: 16976: 16969: 16965: 16952: 16948: 16942: 16939: 16936: 16928: 16925: 16922: 16917: 16914: 16911: 16907: 16901: 16893: 16888: 16884: 16873: 16865: 16860: 16857: 16855: 16848: 16844: 16817: 16813: 16790: 16786: 16777: 16773: 16769: 16764: 16748: 16744: 16721: 16717: 16707: 16694: 16691: 16688: 16685: 16682: 16668: 16664: 16657: 16654: 16649: 16645: 16641: 16636: 16632: 16628: 16623: 16619: 16610: 16606: 16602: 16598: 16594: 16590: 16586: 16582: 16578: 16573: 16571: 16567: 16563: 16559: 16555: 16551: 16547: 16543: 16539: 16535: 16531: 16527: 16523: 16519: 16515: 16511: 16507: 16503: 16499: 16495: 16491: 16487: 16483: 16479: 16475: 16471: 16467: 16463: 16459: 16455: 16451: 16447: 16435: 16428: 16424: 16417: 16413: 16406: 16402: 16392: 16388: 16384: 16381: 16377: 16374: 16373: 16369: 16365: 16362: 16358: 16354: 16351: 16350: 16346: 16343: 16339: 16335: 16331: 16328: 16327: 16323: 16320: 16316: 16312: 16308: 16304: 16301: 16300: 16296: 16291: 16288: 16283: 16281: 16278: 16277: 16274: 16266: 16259: 16255: 16248: 16244: 16239: 16235: 16231: 16223: 16219: 16214: 16210: 16206: 16201: 16198: 16194: 16190: 16186: 16182: 16174: 16170: 16167: 16161: 16157: 16152: 16148: 16143: 16139: 16136: 16130: 16126: 16121: 16117: 16112: 16108: 16102: 16098: 16094: 16090: 16086: 16082: 16077: 16073: 16069: 16065: 16058: 16051: 16047: 16043: 16036: 16033: 16026: 16022: 16020: 16016: 16012: 16008: 16002: 15998: 15993: 15989: 15983: 15979: 15974: 15970: 15964: 15960: 15955: 15951: 15947: 15945: 15941: 15936: 15932: 15928: 15925:decreases as 15924: 15917: 15912: 15908: 15904: 15900: 15895: 15891: 15887: 15883: 15879: 15874: 15870: 15866: 15863: 15860: 15845: 15837: 15825: 15820: 15806: 15802: 15796: 15784: 15777: 15770: 15761: 15757: 15751: 15747: 15737: 15733: 15727: 15723: 15713: 15709: 15703: 15699: 15692: 15687: 15682: 15674: 15670: 15662: 15655: 15651: 15645: 15637: 15632: 15628: 15618: 15614: 15606: 15602: 15596: 15588: 15583: 15579: 15573: 15565: 15560: 15556: 15546: 15542: 15534: 15530: 15524: 15516: 15511: 15507: 15501: 15493: 15488: 15484: 15478: 15470: 15465: 15461: 15451: 15447: 15439: 15435: 15429: 15421: 15416: 15412: 15406: 15398: 15393: 15389: 15383: 15375: 15370: 15366: 15356: 15352: 15344: 15340: 15334: 15326: 15321: 15317: 15311: 15303: 15298: 15294: 15288: 15280: 15275: 15271: 15261: 15257: 15249: 15245: 15239: 15231: 15226: 15222: 15216: 15208: 15203: 15199: 15193: 15179: 15175: 15167: 15163: 15157: 15149: 15144: 15140: 15134: 15120: 15116: 15108: 15104: 15098: 15087: 15077: 15073: 15068: 15052: 15043: 15038: 15034: 15027: 15021: 15018: 15012: 15006: 15000: 14985: 14966: 14958: 14955: 14950: 14942: 14939: 14936: 14931: 14928: 14925: 14921: 14915: 14912: 14909: 14901: 14896: 14892: 14886: 14878: 14876: 14868: 14855: 14852: 14849: 14844: 14836: 14833: 14830: 14825: 14822: 14819: 14815: 14809: 14806: 14803: 14795: 14790: 14786: 14780: 14772: 14770: 14762: 14747: 14743: 14739: 14736: 14731: 14727: 14723: 14720: 14717: 14712: 14709: 14706: 14702: 14696: 14693: 14690: 14686: 14682: 14677: 14674: 14671: 14667: 14661: 14657: 14653: 14651: 14643: 14637: 14625: 14621: 14617: 14613: 14609: 14605: 14600: 14598: 14588: 14586: 14582: 14572: 14569: 14566: 14562: 14559: 14555: 14551: 14548: 14545: 14542: 14541: 14537: 14534: 14531: 14527: 14524: 14520: 14516: 14513: 14510: 14507: 14506: 14502: 14499: 14496: 14493: 14489: 14486: 14483: 14480: 14479: 14475: 14472: 14469: 14466: 14462: 14459: 14456: 14453: 14452: 14449: 14446: 14444: 14441: 14439: 14436: 14434: 14431: 14429: 14426: 14422: 14418: 14415: 14413: 14410: 14409: 14406: 14398: 14396: 14392: 14388: 14384: 14380: 14376: 14358: 14355: 14352: 14348: 14339: 14331: 14328: 14325: 14320: 14317: 14314: 14310: 14301: 14293: 14288: 14284: 14280: 14269: 14265: 14261: 14251: 14249: 14245: 14241: 14223: 14219: 14215: 14212: 14207: 14203: 14199: 14196: 14191: 14187: 14181: 14177: 14173: 14168: 14164: 14158: 14154: 14108: 14106: 14101: 14095: 14087: 14080: 14075: 14060: 14054: 14047: 14041: 14036: 14031: 14023: 14009: 13998: 13991: 13985: 13980: 13973: 13968: 13962: 13952: 13947: 13932: 13926: 13919: 13913: 13908: 13903: 13897: 13894: 13887: 13884: 13878: 13873: 13868: 13860: 13846: 13835: 13828: 13822: 13817: 13810: 13805: 13799: 13778: 13775: 13767: 13763: 13756: 13753: 13748: 13744: 13739: 13736: 13733: 13725: 13721: 13714: 13711: 13706: 13702: 13697: 13694: 13691: 13683: 13679: 13672: 13669: 13664: 13660: 13640: 13637: 13634: 13631: 13628: 13625: 13622: 13619: 13614: 13610: 13606: 13603: 13600: 13595: 13591: 13587: 13584: 13581: 13576: 13572: 13568: 13565: 13562: 13557: 13553: 13549: 13546: 13543: 13538: 13534: 13530: 13527: 13524: 13516: 13512: 13505: 13502: 13497: 13493: 13484: 13481:at powers of 13480: 13464: 13461: 13458: 13455: 13452: 13447: 13443: 13439: 13436: 13431: 13427: 13423: 13420: 13415: 13411: 13407: 13404: 13399: 13395: 13391: 13388: 13383: 13379: 13375: 13372: 13366: 13360: 13357: 13351: 13345: 13342: 13336: 13330: 13310: 13307: 13304: 13301: 13298: 13293: 13289: 13285: 13282: 13277: 13273: 13269: 13266: 13261: 13257: 13253: 13250: 13245: 13241: 13237: 13234: 13229: 13225: 13221: 13218: 13212: 13204: 13200: 13196: 13191: 13187: 13179: 13173: 13170: 13164: 13158: 13139: 13136: 13133: 13130: 13127: 13122: 13118: 13114: 13111: 13106: 13102: 13098: 13095: 13089: 13081: 13071: 13067: 13059: 13053: 13050: 13044: 13036: 13032: 13021: 13017: 13013: 13009: 12992: 12989: 12986: 12983: 12980: 12975: 12971: 12967: 12964: 12959: 12955: 12951: 12948: 12943: 12939: 12935: 12927: 12923: 12919: 12916: 12905: 12901: 12897: 12894: 12883: 12879: 12875: 12872: 12863: 12860: 12857: 12851: 12845: 12839: 12831: 12825: 12818: 12811: 12800: 12798: 12794: 12790: 12786: 12782: 12772: 12765: 12761: 12751: 12749: 12730: 12722: 12709: 12707: 12703: 12699: 12692: 12679: 12677: 12659: 12656: 12651: 12647: 12624: 12620: 12611: 12603: 12592: 12590: 12586: 12581: 12577: 12573: 12555: 12547: 12544: 12541: 12537: 12533: 12526: 12517: 12514: 12511: 12507: 12503: 12494: 12491: 12488: 12484: 12480: 12474: 12469: 12464: 12456: 12444: 12435: 12432: 12429: 12415: 12404: 12397: 12389: 12386: 12383: 12380: 12376: 12370: 12363: 12360: 12357: 12353: 12345: 12341: 12333: 12328: 12323: 12318: 12309: 12306: 12303: 12299: 12293: 12286: 12282: 12274: 12270: 12260: 12256: 12250: 12243: 12239: 12231: 12227: 12220: 12209: 12204: 12200: 12196: 12191: 12176: 12173: 12170: 12166: 12162: 12159: 12154: 12144: 12141: 12138: 12135: 12132: 12128: 12124: 12121: 12118: 12113: 12110: 12107: 12097: 12094: 12091: 12087: 12083: 12078: 12068: 12064: 12041: 12038: 12035: 12031: 12021: 12008: 12005: 12000: 11996: 11990: 11982: 11977: 11974: 11971: 11967: 11961: 11958: 11955: 11947: 11944: 11941: 11936: 11933: 11930: 11927: 11924: 11920: 11914: 11906: 11901: 11898: 11895: 11891: 11881: 11868: 11865: 11861: 11855: 11850: 11846: 11840: 11836: 11830: 11825: 11822: 11819: 11815: 11810: 11804: 11796: 11793: 11790: 11786: 11780: 11777: 11774: 11771: 11768: 11763: 11759: 11753: 11749: 11743: 11738: 11735: 11732: 11728: 11723: 11717: 11709: 11705: 11699: 11696: 11693: 11690: 11687: 11682: 11678: 11672: 11668: 11662: 11657: 11654: 11651: 11647: 11642: 11636: 11628: 11624: 11618: 11615: 11612: 11607: 11603: 11597: 11593: 11587: 11582: 11579: 11576: 11572: 11567: 11537: 11524: 11521: 11517: 11511: 11506: 11502: 11496: 11492: 11486: 11476: 11471: 11468: 11465: 11461: 11456: 11452: 11449: 11446: 11442: 11436: 11433: 11430: 11427: 11424: 11419: 11415: 11409: 11405: 11399: 11389: 11384: 11381: 11378: 11374: 11369: 11365: 11361: 11355: 11352: 11349: 11346: 11343: 11338: 11334: 11328: 11324: 11318: 11308: 11303: 11300: 11297: 11293: 11288: 11284: 11280: 11274: 11271: 11268: 11263: 11259: 11253: 11249: 11243: 11238: 11235: 11232: 11228: 11223: 11213: 11200: 11197: 11193: 11187: 11182: 11178: 11172: 11168: 11162: 11154: 11151: 11148: 11143: 11140: 11137: 11134: 11131: 11126: 11122: 11116: 11112: 11106: 11098: 11093: 11090: 11087: 11084: 11081: 11076: 11072: 11066: 11062: 11056: 11048: 11043: 11040: 11037: 11032: 11028: 11022: 11018: 11013: 11007: 11002: 10999: 10996: 10992: 10983: 10979: 10974: 10957: 10954: 10949: 10944: 10940: 10934: 10930: 10924: 10916: 10913: 10910: 10905: 10902: 10899: 10896: 10893: 10888: 10884: 10878: 10874: 10868: 10860: 10855: 10852: 10849: 10846: 10843: 10838: 10834: 10828: 10824: 10818: 10810: 10805: 10802: 10799: 10794: 10790: 10784: 10780: 10771: 10768: 10763: 10760: 10755: 10751: 10745: 10742: 10739: 10734: 10730: 10724: 10720: 10714: 10706: 10703: 10700: 10695: 10692: 10687: 10683: 10677: 10674: 10671: 10666: 10662: 10656: 10652: 10646: 10638: 10633: 10630: 10625: 10621: 10615: 10612: 10609: 10604: 10600: 10594: 10590: 10584: 10576: 10571: 10568: 10565: 10560: 10556: 10550: 10546: 10537: 10534: 10530: 10524: 10521: 10516: 10512: 10506: 10498: 10495: 10492: 10487: 10484: 10479: 10475: 10469: 10461: 10456: 10453: 10448: 10444: 10438: 10430: 10427: 10423: 10417: 10414: 10411: 10406: 10402: 10396: 10392: 10383: 10380: 10372: 10369: 10364: 10360: 10348: 10345: 10342: 10337: 10333: 10327: 10323: 10295: 10292: 10289: 10284: 10280: 10274: 10270: 10249: 10246: 10243: 10240: 10237: 10217: 10208: 10195: 10192: 10184: 10181: 10176: 10172: 10145: 10142: 10139: 10136: 10133: 10130: 10122: 10118: 10114: 10109: 10106: 10101: 10097: 10093: 10090: 10065: 10062: 10057: 10053: 10049: 10046: 10024: 10021: 10016: 10012: 10001: 9996:The zeros of 9994: 9979: 9975: 9969: 9961: 9958: 9955: 9950: 9946: 9940: 9932: 9927: 9923: 9917: 9909: 9906: 9903: 9895: 9891: 9887: 9884: 9881: 9873: 9868: 9865: 9862: 9858: 9854: 9848: 9832: 9827: 9822: 9820: 9807: 9793: 9773: 9770: 9761: 9747: 9744: 9741: 9733: 9726: 9721: 9720: 9714: 9706: 9691: 9683: 9680: 9677: 9673: 9665: 9656: 9652: 9642: 9638: 9631: 9626: 9621: 9613: 9609: 9601: 9592: 9588: 9578: 9574: 9567: 9560: 9552: 9549: 9546: 9541: 9537: 9531: 9524: 9521: 9518: 9513: 9509: 9501: 9498: 9495: 9490: 9486: 9478: 9473: 9468: 9463: 9454: 9449: 9445: 9439: 9432: 9427: 9423: 9415: 9410: 9406: 9396: 9391: 9387: 9381: 9374: 9369: 9365: 9357: 9352: 9348: 9341: 9331: 9324: 9317: 9310: 9305: 9301: 9297: 9291: 9275: 9270: 9266: 9262: 9257: 9245: 9241: 9236: 9229: 9222: 9218: 9214: 9211: 9207: 9203: 9196: 9192: 9181: 9177: 9164: 9149: 9144: 9140: 9134: 9130: 9124: 9119: 9116: 9113: 9109: 9105: 9100: 9096: 9086: 9069: 9065: 9060: 9056: 9051: 9047: 9040: 9033: 9029: 9024: 9020: 9015: 9011: 9002: 8996: 8992: 8986: 8976: 8972: 8956: 8952: 8944:has roots at 8928: 8922: 8902: 8899: 8891: 8887: 8880: 8856: 8853: 8850: 8847: 8844: 8841: 8838: 8835: 8832: 8829: 8826: 8822: 8815: 8811: 8805: 8800: 8796: 8792: 8783: 8779: 8774: 8768: 8763: 8760: 8757: 8753: 8749: 8747: 8734: 8730: 8723: 8720: 8718: 8705: 8701: 8694: 8691: 8688: 8685: 8677: 8673: 8666: 8663: 8655: 8651: 8644: 8641: 8633: 8629: 8622: 8619: 8617: 8610: 8606: 8592: 8588: 8570: 8567: 8564: 8560: 8556: 8551: 8547: 8532: 8530: 8526: 8522: 8518: 8514: 8510: 8506: 8496: 8489: 8484: 8467: 8463: 8458: 8450: 8446: 8441: 8435: 8430: 8427: 8424: 8420: 8416: 8410: 8404: 8396: 8392: 8385: 8381: 8377: 8369: 8354: 8350: 8344: 8340: 8334: 8331: 8328: 8323: 8320: 8317: 8313: 8309: 8303: 8297: 8275: 8269: 8266: 8260: 8254: 8251: 8245: 8239: 8231: 8227: 8223: 8219: 8214: 8201: 8198: 8195: 8192: 8189: 8186: 8183: 8180: 8177: 8174: 8171: 8165: 8162: 8159: 8151: 8147: 8140: 8120: 8117: 8109: 8105: 8101: 8098: 8082: 8076: 8073: 8065: 8061: 8057: 8054: 8038: 8032: 8024: 8020: 8016: 8012: 8007: 8005: 7989: 7981: 7977: 7973: 7970: 7962: 7959: 7956: 7951: 7948: 7945: 7941: 7937: 7931: 7925: 7917: 7913: 7909: 7905: 7900: 7885: 7881: 7875: 7871: 7865: 7862: 7859: 7854: 7851: 7848: 7844: 7840: 7834: 7828: 7820: 7816: 7795: 7792: 7789: 7785: 7781: 7778: 7775: 7770: 7766: 7762: 7759: 7756: 7751: 7747: 7729: 7727: 7723: 7716: 7706: 7698: 7696: 7691: 7687: 7678: 7676: 7672: 7656: 7648: 7630: 7626: 7622: 7617: 7613: 7584: 7580: 7573: 7570: 7567: 7564: 7556: 7552: 7545: 7519: 7516: 7510: 7504: 7476: 7473: 7470: 7466: 7459: 7456: 7453: 7450: 7442: 7438: 7431: 7408: 7388: 7363: 7355: 7352: 7329: 7326: 7323: 7320: 7317: 7314: 7311: 7305: 7302: 7280: 7276: 7255: 7235: 7228:of the field 7227: 7211: 7202: 7197: 7195: 7191: 7187: 7182: 7180: 7175: 7158: 7155: 7152: 7146: 7140: 7137: 7134: 7111: 7108: 7105: 7085: 7082: 7079: 7071: 7055: 7052: 7049: 7029: 7021: 7020:byte-oriented 7005: 7002: 6999: 6996: 6991: 6987: 6983: 6980: 6960: 6957: 6952: 6948: 6944: 6941: 6921: 6899: 6895: 6874: 6865: 6851: 6831: 6808: 6805: 6800: 6796: 6791: 6786: 6783: 6761: 6753: 6750: 6747: 6741: 6738: 6735: 6730: 6726: 6702: 6699: 6690: 6681: 6678: 6673: 6670: 6648: 6645: 6642: 6632: 6628: 6624: 6621: 6613: 6608: 6604: 6592: 6589: 6578: 6573: 6568: 6565: 6562: 6559: 6556: 6552: 6546: 6543: 6536: 6533: 6528: 6523: 6519: 6496: 6493: 6490: 6480: 6476: 6472: 6469: 6461: 6456: 6452: 6440: 6437: 6426: 6421: 6416: 6413: 6410: 6407: 6404: 6400: 6394: 6391: 6383: 6380: 6375: 6371: 6364: 6361: 6358: 6354: 6348: 6343: 6339: 6330: 6326: 6317: 6313: 6299: 6279: 6270: 6266: 6262: 6258: 6252: 6248: 6244: 6241: 6225: 6221: 6214: 6211: 6208: 6185: 6182: 6179: 6170: 6168: 6164: 6160: 6156: 6140: 6137: 6134: 6131: 6128: 6125: 6116: 6108: 6104: 6101: 6097: 6093: 6089: 6079: 6066: 6059: 6053: 6048: 6044: 6041: 6035: 6027: 6023: 6019: 6013: 6005: 6001: 5997: 5991: 5983: 5979: 5975: 5970: 5966: 5962: 5956: 5950: 5947: 5941: 5935: 5912: 5906: 5861: 5855: 5846: 5833: 5826: 5818: 5814: 5810: 5805: 5801: 5797: 5791: 5785: 5782: 5776: 5770: 5747: 5741: 5721: 5698: 5692: 5670: 5666: 5662: 5656: 5650: 5628: 5624: 5620: 5615: 5611: 5607: 5604: 5601: 5596: 5593: 5590: 5586: 5582: 5577: 5574: 5571: 5567: 5546: 5543: 5540: 5531: 5518: 5512: 5506: 5488: 5484: 5480: 5474: 5468: 5465: 5459: 5451: 5447: 5423: 5415: 5411: 5390: 5367: 5361: 5341: 5338: 5335: 5332: 5329: 5307: 5303: 5279: 5273: 5264: 5247: 5241: 5218: 5212: 5189: 5183: 5160: 5154: 5131: 5125: 5102: 5096: 5073: 5067: 5047: 5024: 5018: 4995: 4989: 4983: 4977: 4974: 4968: 4962: 4954: 4944: 4931: 4927: 4921: 4918: 4915: 4911: 4907: 4904: 4901: 4896: 4892: 4888: 4883: 4879: 4868: 4864: 4858: 4854: 4848: 4843: 4840: 4837: 4833: 4829: 4823: 4817: 4804: 4798: 4794: 4790: 4787: 4784: 4779: 4775: 4771: 4766: 4762: 4757: 4752: 4748: 4735: 4721: 4718: 4715: 4706: 4691: 4688: 4685: 4681: 4677: 4672: 4669: 4666: 4663: 4660: 4656: 4650: 4647: 4644: 4641: 4638: 4634: 4630: 4627: 4624: 4621: 4616: 4612: 4608: 4603: 4599: 4595: 4591: 4585: 4582: 4579: 4576: 4573: 4570: 4567: 4563: 4559: 4556: 4552: 4548: 4544: 4538: 4535: 4532: 4528: 4524: 4521: 4517: 4512: 4506: 4502: 4498: 4495: 4491: 4487: 4481: 4475: 4456: 4433: 4427: 4407: 4404: 4401: 4378: 4372: 4365: 4349: 4346: 4343: 4320: 4314: 4291: 4285: 4262: 4256: 4236: 4233: 4230: 4227: 4224: 4204: 4201: 4198: 4175: 4169: 4146: 4140: 4126: 4124: 4118: 4100: 4097: 4094: 4090: 4086: 4083: 4080: 4075: 4071: 4067: 4064: 4061: 4058: 4052: 4047: 4044: 4041: 4037: 4033: 4030: 4027: 4022: 4018: 3996: 3992: 3988: 3983: 3979: 3970: 3966: 3962: 3958: 3953: 3938: 3927: 3924: 3921: 3917: 3908: 3904: 3896: 3884: 3880: 3871: 3867: 3854: 3850: 3841: 3837: 3830: 3825: 3819: 3813: 3791: 3787: 3778: 3768: 3752: 3748: 3725: 3722: 3719: 3715: 3711: 3708: 3705: 3700: 3696: 3670: 3667: 3664: 3660: 3651: 3647: 3643: 3640: 3637: 3629: 3625: 3616: 3612: 3591: 3583: 3578: 3563: 3552: 3549: 3546: 3542: 3533: 3529: 3521: 3509: 3505: 3496: 3492: 3479: 3475: 3466: 3462: 3455: 3450: 3444: 3438: 3430: 3414: 3411: 3408: 3404: 3400: 3397: 3394: 3389: 3385: 3376: 3360: 3338: 3334: 3313: 3307: 3304: 3301: 3298: 3295: 3292: 3289: 3283: 3280: 3270: 3266: 3262: 3254: 3250: 3241: 3237: 3216: 3194: 3190: 3167: 3163: 3154: 3144: 3130: 3122: 3118: 3102: 3094: 3089: 3074: 3066: 3063: 3060: 3056: 3048: 3039: 3035: 3025: 3021: 3014: 3007: 2999: 2996: 2993: 2988: 2985: 2982: 2978: 2972: 2965: 2960: 2957: 2954: 2950: 2942: 2939: 2936: 2932: 2926: 2919: 2914: 2909: 2904: 2899: 2890: 2887: 2884: 2879: 2875: 2869: 2862: 2857: 2853: 2845: 2841: 2835: 2826: 2823: 2820: 2815: 2811: 2805: 2798: 2793: 2789: 2781: 2777: 2771: 2765: 2760: 2757: 2754: 2751: 2745: 2739: 2719: 2699: 2679: 2676: 2673: 2653: 2650: 2647: 2641: 2635: 2627: 2611: 2589: 2578: 2575: 2572: 2568: 2559: 2555: 2547: 2535: 2531: 2522: 2518: 2505: 2501: 2492: 2488: 2481: 2476: 2470: 2464: 2442: 2438: 2429: 2425: 2421: 2418: 2398: 2391:of the field 2376: 2373: 2370: 2366: 2362: 2359: 2356: 2351: 2347: 2326: 2304: 2300: 2279: 2259: 2253: 2249: 2243: 2239: 2233: 2230: 2227: 2222: 2219: 2216: 2212: 2208: 2202: 2194: 2190: 2167: 2163: 2140: 2136: 2132: 2124: 2121: 2118: 2114: 2110: 2107: 2104: 2099: 2095: 2088: 2085: 2077: 2067: 2065: 2061: 2057: 2039: 2035: 2031: 2028: 2025: 2020: 2016: 2007: 2003: 1999: 1995: 1991: 1987: 1969: 1965: 1956: 1951: 1949: 1933: 1929: 1925: 1922: 1919: 1916: 1913: 1910: 1907: 1899: 1895: 1879: 1875: 1871: 1868: 1865: 1845: 1842: 1839: 1836: 1833: 1829: 1825: 1822: 1819: 1816: 1813: 1810: 1807: 1803: 1799: 1796: 1793: 1789: 1785: 1782: 1779: 1776: 1773: 1769: 1765: 1762: 1759: 1739: 1736: 1733: 1730: 1727: 1724: 1721: 1713: 1697: 1694: 1691: 1671: 1668: 1665: 1662: 1659: 1656: 1650: 1647: 1644: 1638: 1635: 1615: 1612: 1609: 1589: 1581: 1565: 1553: 1550: 1542: 1534: 1510: 1506: 1499: 1496: 1493: 1490: 1482: 1478: 1471: 1468: 1460: 1456: 1449: 1431: 1395: 1393: 1389: 1385: 1381: 1377: 1373: 1369: 1365: 1361: 1357: 1353: 1350:− 1}, {0, 1, 1349: 1345: 1342:of the field 1327: 1323: 1319: 1316: 1313: 1308: 1304: 1295: 1291: 1287: 1271: 1251: 1243: 1227: 1207: 1199: 1189: 1175: 1172: 1169: 1166: 1163: 1143: 1140: 1137: 1115: 1112: 1107: 1104: 1097: 1093: 1077: 1057: 1037: 1030: 1014: 1011: 1008: 1005: 1002: 994: 979: 972: 956: 949: 933: 925: 910: 908: 905: 902: 901: 897: 895: 892: 889: 888: 884: 881: 878: 877: 873: 870: 867: 866: 862: 859: 856: 855: 851: 848: 845: 844: 840: 838: 835: 832: 831: 827: 824: 821: 818: 817: 813: 810: 807: 804: 803: 799: 796: 793: 792: 788: 785: 782: 781: 775: 773: 768: 766: 762: 758: 754: 750: 746: 741: 739: 735: 733: 729: 724: 722: 714: 709: 700: 698: 695:as a form of 694: 690: 686: 681: 679: 675: 671: 667: 662: 660: 656: 652: 648: 644: 640: 636: 632: 622: 620: 615: 613: 609: 605: 601: 597: 587: 585: 581: 577: 572: 569: 565: 563: 559: 556: 555:convolutional 552: 548: 544: 539: 526: 524: 519: 517: 516: 510: 508: 503: 501: 497: 493: 490: 489:convolutional 486: 482: 478: 474: 470: 466: 462: 458: 454: 446: 442: 440: 435: 433: 429: 425: 420: 418: 414: 409: 406:developed by 405: 400: 397: 393: 389: 385: 381: 377: 373: 363: 361: 356: 344: 339: 331: 325: 319: 308: − 299: 294: 292: 288: 284: 280: 276: 272: 268: 264: 260: 256: 252: 248: 244: 240: 236: 232: 228: 217: 212: 210: 205: 203: 198: 197: 194: 191: 187: 182: 179: 175: 171: 167: 162: 157: 153: 151: 147: 143: 139: 133: 129: 125: 121: 118: 116: 115:Alphabet size 112: 108: 104: 101: 99: 95: 92: 89: 87: 83: 80: 77: 75: 71: 66: 62: 59: 55: 50: 47: 43: 40: 36: 31: 19: 22501: 22476: 22422:, Springer, 22418: 22395: 22382:the original 22376: 22359: 22353: 22341: 22337: 22312: 22292: 22276: 22259: 22255: 22232: 22228: 22199: 22171: 22155: 22151: 22133:, retrieved 22126:the original 22117: 22095:. Retrieved 22083: 22070: 22043: 22037: 22031: 22021:, retrieved 22006: 21999: 21962: 21952: 21915: 21905: 21879: 21873: 21867: 21858: 21850: 21846: 21837: 21830:compact disc 21822:, retrieved 21810: 21806: 21793: 21782:. Retrieved 21768: 21755: 21730: 21726: 21683: 21677: 21655: 21646: 21633: 21607:(1): 87–99. 21604: 21600: 21590: 21562: 21555: 21538: 21532: 21507: 21503: 21497: 21486: 21462: 21433:Chien search 21404: 21396: 21389: 21385: 21378: 21374: 21372:Recalculate 21371: 21365: 21361: 21357: 21353: 21349: 21345: 21341: 21337: 21329: 21325: 21321: 21317: 21309: 21305: 21301: 21297: 21293: 21289: 21282: 21278: 21274: 21270: 21266: 21262: 21260: 21254: 21250: 21243: 21239: 21235: 21227: 21223: 21219: 21215: 21208: 21204: 21200: 21182:152 x + 237 21146: 21138: 21131: 21051: 21047: 20880: 20872: 20861: 20853: 20843:Recalculate 20842: 20836: 20832: 20828: 20824: 20820: 20816: 20812: 20808: 20800: 20796: 20792: 20788: 20780: 20776: 20772: 20768: 20764: 20760: 20295: 20288: 19825: 19601: 19596: 19594: 19519: 19516: 19510: 19506: 19502: 19497: 19491: 19486: 19480: 19476: 19472: 19468: 19463: 19457: 19450: 19445: 19441: 19439: 19370: 19188: 19179: 18614:% magnitudes 17666:% Example: 17644: 17339: 17300: 17289: 17285: 17278: 17219: 17212: 17208: 17200: 17194: 17188: 17184: 17179: 17175: 17171: 17165: 17155: 17151: 17147: 17143: 17139: 17135: 17131: 17127: 17123: 17119: 17117: 16775: 16771: 16767: 16765: 16708: 16608: 16604: 16600: 16596: 16592: 16588: 16584: 16580: 16576: 16574: 16569: 16565: 16561: 16557: 16553: 16549: 16545: 16541: 16537: 16533: 16529: 16525: 16521: 16517: 16513: 16509: 16505: 16501: 16497: 16493: 16489: 16485: 16481: 16477: 16473: 16469: 16465: 16461: 16457: 16453: 16449: 16445: 16443: 16433: 16432:/ 544 = 546 16426: 16422: 16415: 16411: 16410:/ 544 = 329 16404: 16400: 16390: 16386: 16379: 16367: 16360: 16356: 16341: 16337: 16333: 16318: 16314: 16310: 16306: 16294: 16286: 16279: 16272: 16257: 16253: 16252: 16246: 16242: 16237: 16233: 16229: 16221: 16217: 16212: 16208: 16204: 16196: 16192: 16188: 16184: 16180: 16178: 16172: 16168: 16165: 16159: 16155: 16150: 16146: 16141: 16137: 16134: 16128: 16124: 16119: 16115: 16110: 16106: 16100: 16096: 16092: 16088: 16084: 16080: 16075: 16071: 16067: 16063: 16056: 16049: 16045: 16041: 16034: 16031: 16024: 16018: 16014: 16010: 16006: 16005: 16000: 15996: 15991: 15987: 15981: 15977: 15972: 15968: 15962: 15958: 15953: 15949: 15943: 15939: 15934: 15930: 15926: 15922: 15920: 15915: 15910: 15906: 15902: 15898: 15893: 15889: 15885: 15881: 15877: 15872: 15868: 15864: 15861: 15075: 15071: 15069: 14986: 14623: 14619: 14615: 14611: 14607: 14603: 14601: 14594: 14584: 14580: 14578: 14564: 14557: 14553: 14529: 14522: 14518: 14491: 14464: 14447: 14442: 14437: 14432: 14427: 14420: 14416: 14411: 14404: 14394: 14390: 14386: 14378: 14374: 14267: 14263: 14257: 14247: 14243: 14239: 14146:Subtracting 14145: 14102: 14099: 14093: 14085: 14078: 13948: 13482: 13478: 13019: 13015: 13011: 13007: 12823: 12816: 12809: 12806: 12796: 12792: 12788: 12784: 12774: 12767: 12763: 12759: 12757: 12744: 12717: 12715: 12694: 12687: 12685: 12676:Chien search 12606: 12601: 12598: 12584: 12579: 12575: 12571: 12207: 12202: 12198: 12194: 12193:Recall that 12192: 12023:Subtracting 12022: 11882: 11538: 11214: 10981: 10977: 10975: 10209: 9999: 9995: 9830: 9825: 9823: 9815: 9813: 9762: 9728: 9725:erasure code 9722: 9716: 9709: 9707: 9326: 9319: 9312: 9308: 9303: 9299: 9295: 9292: 9165: 9087: 8997: 8995:error values 8994: 8987: 8984: 8982: 8973: 8590: 8586: 8538: 8528: 8524: 8520: 8516: 8512: 8508: 8498: 8491: 8487: 8485: 8394: 8387: 8383: 8379: 8372: 8371:Coefficient 8370: 8229: 8225: 8221: 8217: 8215: 8022: 8018: 8014: 8010: 8008: 8003: 7915: 7911: 7907: 7903: 7901: 7818: 7814: 7735: 7721: 7718: 7704: 7692: 7688: 7684: 7198: 7193: 7183: 7176: 7069: 6866: 6327:channel for 6322: 6268: 6264: 6260: 6256: 6171: 6158: 6154: 6105:and minimum 6102: 6095: 6091: 6085: 5847: 5532: 5265: 4950: 4736: 4707: 4132: 4119: 3968: 3964: 3960: 3956: 3954: 3774: 3579: 3431: 3150: 3090: 2073: 2059: 2055: 2005: 2001: 1997: 1994:coefficients 1993: 1989: 1954: 1952: 1897: 1579: 1396: 1391: 1383: 1379: 1375: 1371: 1367: 1363: 1359: 1355: 1351: 1347: 1343: 1293: 1289: 1285: 1241: 1195: 1070:, and thus, 1029:finite field 992: 948:block length 921: 890:2004–present 879:1996–present 846:1977–present 833:1977–present 794:1958–present 769: 742: 736: 728:concatenated 725: 718: 687:systems and 682: 677: 676:is usually 2 673: 669: 665: 663: 658: 654: 650: 646: 642: 628: 616: 593: 573: 570: 566: 543:compact disc 540: 537: 534:Data storage 529:Applications 520: 513: 511: 504: 471:devices and 463:, where two 461:compact disc 450: 436: 428:James Massey 421: 417:cyclic codes 401: 391: 387: 369: 357: 342: 330:erasure code 323: 317: 298:finite-field 295: 226: 225: 177: 155: 141: 137: 131: 127: 123: 119: 106: 102: 90: 78: 74:Block length 18:Reed–Solomon 22493:Dave Forney 22484:, TM-102162 22303:, AD0669824 21438:Cyclic code 21394:to correct 20869:Gao decoder 20851:to correct 17653:= rsDecoder 17552:k, n, alpha 17312:turbo codes 17304:demodulator 17222:Madhu Sudan 16066: := 0 16062: := 1 16055: := 0 9824:Define the 8133:Therefore, 7732:Formulation 6240:demodulator 4123:Gao decoder 3117:linear code 1092:prime power 894:Turbo codes 789:Mission(s) 772:turbo codes 635:Vandermonde 621:symbology. 558:interleaver 465:interleaved 263:Data Matrix 38:Named after 22558:Categories 22097:2017-06-07 22023:2024-02-08 22017:2304.09445 21972:2304.01403 21925:2206.05256 21824:2009-09-28 21784:2019-02-01 21660:IEEE Press 21478:References 18983:decoded_gf 18932:decoded_gf 18701:size_error 18680:size_error 18617:size_error 18344:max_errors 17672:max_errors 17519:msg_padded 17498:msg_padded 17459:msg_padded 16575:Transform 16070:degree of 12716:Calculate 8873:Note that 7695:puncturing 6090:of length 6082:Properties 4394:of degree 4191:of degree 3582:systematic 3229:such that 3153:systematic 1362:}, or for 1090:must be a 907:LDPC codes 612:Aztec Code 604:Datamatrix 492:inner code 184:Properties 164:Algorithms 22338:SIAM News 22311:(1984) , 22135:April 21, 22048:CiteSeerX 21884:CiteSeerX 21702:557445046 21640:, Clemson 21547:859669631 21068:− 20935:− 20909:∏ 20897:− 19787:− 19658:− 19556:− 19301:− 19099:prim_poly 18959:errors_gf 18923:prim_poly 18899:errors_gf 18884:error_mag 18869:error_pos 18848:prim_poly 18821:prim_poly 18794:error_mag 18734:error_pos 18641:Syndromes 18629:error_pos 18593:error_mag 18575:orig_vals 18563:error_pos 18512:error_pos 18401:remainder 18326:remainder 18119:prim_poly 18077:prim_poly 18053:Syndromes 18005:prim_poly 17927:error_mag 17918:error_pos 17900:orig_vals 17882:Syndromes 17852:Syndromes 17795:prim_poly 17705:orig_vals 17525:remainder 17477:prim_poly 17414:prim_poly 17356:rsEncoder 17220:In 1999, 17084:for 17070:− 17053:Λ 17046:⋯ 17035:− 17018:Λ 17006:− 16989:Λ 16982:− 16940:− 16933:Λ 16926:⋯ 16915:− 16898:Λ 16870:Λ 16861:− 16692:≤ 16686:≤ 16679:for 16665:α 16524:). Since 16154: := 16123: := 16095: := 16087: := 16040: := 16030: := 15946:/2, then 15834:Ω 15817:Ω 15793:Ω 15642:Λ 15593:Λ 15570:Λ 15521:Λ 15498:Λ 15475:Λ 15426:Λ 15403:Λ 15380:Λ 15331:Λ 15308:Λ 15285:Λ 15236:Λ 15213:Λ 15190:Λ 15154:Λ 15131:Λ 15095:Λ 15047:Ω 14995:Λ 14963:Ω 14947:Ω 14940:⋯ 14929:− 14913:− 14906:Ω 14883:Ω 14863:Ω 14841:Λ 14834:⋯ 14823:− 14807:− 14800:Λ 14777:Λ 14757:Λ 14721:⋯ 14710:− 14694:− 14675:− 14614:), and Ω( 14356:− 14336:Λ 14329:⋯ 14318:− 14298:Λ 14278:Δ 14091:= 562 = 3 14020:Λ 14006:Λ 13895:− 13885:− 13857:Λ 13843:Λ 13626:⋅ 13607:⋅ 13588:⋅ 13569:⋅ 13550:⋅ 13531:⋅ 13197:− 12920:− 12898:− 12876:− 12861:− 12758:Finally, 12731:α 12657:− 12589:Gill n.d. 12548:ν 12542:ν 12534:− 12527:⋮ 12512:ν 12504:− 12489:ν 12481:− 12453:Λ 12445:⋮ 12433:− 12430:ν 12426:Λ 12416:ν 12412:Λ 12387:− 12384:ν 12371:⋯ 12358:ν 12346:ν 12334:⋮ 12329:⋱ 12324:⋮ 12319:⋮ 12304:ν 12294:⋯ 12261:ν 12251:⋯ 12177:ν 12163:− 12151:Λ 12142:− 12139:ν 12122:⋯ 12111:− 12108:ν 12104:Λ 12079:ν 12075:Λ 12042:ν 11991:ν 11987:Λ 11959:− 11956:ν 11952:Λ 11945:⋯ 11934:− 11931:ν 11911:Λ 11902:ν 11831:ν 11816:∑ 11805:ν 11801:Λ 11794:⋯ 11778:− 11775:ν 11744:ν 11729:∑ 11714:Λ 11697:− 11694:ν 11663:ν 11648:∑ 11633:Λ 11619:ν 11588:ν 11573:∑ 11547:Λ 11487:ν 11483:Λ 11477:ν 11462:∑ 11450:⋯ 11434:− 11431:ν 11396:Λ 11390:ν 11375:∑ 11353:− 11350:ν 11315:Λ 11309:ν 11294:∑ 11275:ν 11244:ν 11229:∑ 11163:ν 11159:Λ 11152:⋯ 11141:− 11138:ν 11103:Λ 11091:− 11088:ν 11053:Λ 11044:ν 11008:ν 10993:∑ 10925:ν 10921:Λ 10914:⋯ 10903:− 10900:ν 10865:Λ 10853:− 10850:ν 10815:Λ 10806:ν 10764:ν 10761:− 10746:ν 10715:ν 10711:Λ 10704:⋯ 10693:− 10678:ν 10643:Λ 10631:− 10616:ν 10581:Λ 10572:ν 10525:ν 10522:− 10507:ν 10503:Λ 10496:⋯ 10485:− 10466:Λ 10454:− 10435:Λ 10418:ν 10370:− 10354:Λ 10349:ν 10296:ν 10250:ν 10247:≤ 10241:≤ 10182:− 10166:Λ 10137:− 10115:⋅ 10107:− 10094:− 10063:− 10022:− 9980:ν 9970:ν 9966:Λ 9959:⋯ 9937:Λ 9914:Λ 9885:− 9874:ν 9859:∏ 9843:Λ 9745:− 9681:− 9666:⋮ 9614:ν 9602:⋮ 9550:− 9542:ν 9532:⋯ 9522:− 9499:− 9479:⋮ 9474:⋱ 9469:⋮ 9464:⋮ 9450:ν 9440:⋯ 9392:ν 9382:⋯ 9237:α 9215:∗ 9208:α 9178:α 9125:ν 9110:∑ 9025:α 8953:α 8888:α 8854:− 8845:… 8797:α 8769:ν 8754:∑ 8731:α 8702:α 8674:α 8652:α 8630:α 8568:− 8561:α 8557:… 8548:α 8436:ν 8421:∑ 8332:− 8314:∑ 8199:− 8190:… 8148:α 8106:α 8102:− 8062:α 8058:− 7978:α 7974:− 7960:− 7942:∏ 7863:− 7845:∑ 7793:− 7779:… 7760:… 7627:α 7614:α 7581:α 7568:… 7553:α 7517:α 7508:↦ 7474:− 7467:α 7454:… 7439:α 7327:− 7318:… 7306:∈ 7277:α 7212:α 6997:− 6958:− 6806:⁡ 6751:− 6742:− 6700:− 6649:ℓ 6646:− 6625:− 6614:ℓ 6593:ℓ 6579:ℓ 6557:ℓ 6553:∑ 6529:≈ 6497:ℓ 6494:− 6473:− 6462:ℓ 6441:ℓ 6427:ℓ 6405:ℓ 6401:∑ 6381:− 6362:− 6349:≈ 6212:− 6183:− 6132:− 6100:dimension 6042:≡ 6020:− 5998:≡ 5976:− 5963:⋅ 5948:≡ 5811:− 5798:⋅ 5663:⋅ 5605:… 5594:− 5575:− 5544:− 5481:⋅ 5339:− 4919:− 4912:α 4905:… 4893:α 4880:α 4834:∑ 4788:… 4689:− 4670:− 4664:− 4648:− 4642:− 4628:⋯ 4583:− 4577:− 4564:α 4560:− 4549:⋯ 4529:α 4525:− 4503:α 4499:− 4457:α 4405:− 4362:, with a 4347:− 4234:− 4228:≤ 4202:− 4098:− 4091:α 4084:… 4072:α 4065:α 4045:− 4031:… 3993:α 3925:− 3897:⋯ 3723:− 3709:… 3668:− 3641:… 3604:entries, 3550:− 3522:⋯ 3412:− 3398:… 3305:− 3296:… 3284:∈ 3064:− 3049:⋮ 2997:− 2986:− 2973:… 2958:− 2940:− 2920:⋮ 2915:⋱ 2910:⋮ 2905:⋮ 2900:⋮ 2888:− 2870:… 2824:− 2806:… 2677:× 2576:− 2548:⋯ 2435:→ 2374:− 2360:… 2231:− 2213:∑ 2133:∈ 2122:− 2108:… 2029:… 1917:≤ 1908:δ 1843:− 1837:∼ 1817:− 1783:− 1760:δ 1731:− 1695:− 1663:− 1648:− 1639:− 1613:− 1494:… 1317:… 1173:− 1050:of order 1012:≤ 903:est. 2009 868:1989–2003 857:1989–2003 819:1969–1975 805:1968–1978 404:BCH codes 279:broadcast 243:MiniDiscs 174:Euclidean 57:Hierarchy 22393:(1977), 22332:(1993), 22301:45195002 22291:(1967), 22191:(1969), 22088:Archived 21815:archived 21778:Archived 21582:45727875 21418:BCH code 21412:See also 21360:) = 003 21324:) = 001 21296:) = 003 20831:) = 003 20795:) = 001 20767:) = 003 19475:) = 003 19412:, where 19111:function 19001:function 18530:evalPoly 18458:evalPoly 18233:quotient 18200:quotient 18188:quotient 17650:function 17540:function 17349:function 17187:− 16736:through 16187:) and Ω( 15074:= 6 and 14626:errors: 10976:Sum for 8915:because 7728:(1961). 7605:; since 7345:, where 7190:inverted 6267:− 6251:MDS code 6247:erasures 2692:-matrix 1858:, where 1712:distance 1580:distinct 969:, and a 924:alphabet 874:Galileo 863:Voyager 841:Voyager 633:-RS and 600:MaxiCode 590:Bar code 576:parchive 396:BCH code 259:QR codes 150:Notation 130: ( 98:Distance 22491:by Dr. 22218:9003708 21747:9289140 21524:2098821 21504:J. SIAM 21261:divide 21050:= 1 to 20877:Example 19436:Example 18977:decoded 18938:encoded 18569:decoded 18557:isempty 18464:polyval 18107:size_r0 18029:size_r0 17894:decoded 17876:isempty 17819:encoded 17813:polyval 17711:encoded 17641:Decoder 17547:genpoly 17513:encoded 17429:genpoly 17351:encoded 17336:Encoder 17288:− 17211:− 16516:), and 16492:), and 16269:Example 15980:) = −Q( 15942:) < 14602:Define 14401:Example 14143:) = 122 14128:) = 074 12803:Example 12700:in the 10980:= 1 to 8397:, then 7681:Remarks 7042:, with 6157:, 6098:) with 2008:points 1992:as the 1378:, ..., 1358:, ..., 825:(32, 6) 797:Uncoded 757:Cassini 749:Galileo 645:, 619:PostBar 608:QR Code 596:PDF-417 366:History 257:discs, 255:Blu-ray 22426: 22403: 22319: 22299: 22216: 22178: 22050: 21987: 21940: 21886: 21745: 21700: 21690: 21666: 21580: 21570: 21545: 21522: 21383:where 21364:+ 002 21328:+ 924 21308:+ 007 21304:+ 009 21300:+ 916 21269:) and 21253:+ 176 21249:= 708 21226:+ 311 21222:+ 798 21218:+ 086 21214:= 266 20847:where 20835:+ 002 20799:+ 924 20779:+ 007 20775:+ 009 20771:+ 916 20289:Using 19479:+ 002 19134:length 18911:errors 18863:errors 18623:length 18602:return 18281:length 18206:length 18167:deconv 18035:length 17957:return 17726:errors 17492:deconv 17342:MATLAB 17215:) / 2⌋ 16603:) and 16468:) and 16414:+ 821 16393:+ 544 16389:+ 704 16382:+ 596 16370:+ 396 16363:+ 024 16359:+ 676 16344:+ 732 16340:+ 637 16336:+ 762 16321:+ 000 16317:+ 000 16313:+ 000 16309:+ 000 16091:+ 1 16013:) and 15999:) = Ω( 15961:) = Λ( 14618:) for 14556:+ 821 14521:+ 173 14377:) and 14345: 14307: 14139:)/Λ'(x 14135:= −Ω(x 14124:)/Λ'(x 14120:= −Ω(x 13949:Using 13014:) = 3 12830:PDF417 9044: 8169: 8009:Since 8002:where 7671:cyclic 7201:cyclic 7179:bursts 6663:where 6094:(over 5502: 5494: 4217:where 2002:values 1370:, {1, 631:Cauchy 610:, and 580:USENET 500:DVB-S2 291:RAID 6 178:et al. 136:Often 134:prime) 22506:(PDF) 22450:(CMU) 22281:(PDF) 22214:S2CID 22196:(PDF) 22148:(PDF) 22129:(PDF) 22122:(PDF) 22091:(PDF) 22080:(PDF) 22012:arXiv 21967:arXiv 21920:arXiv 21818:(PDF) 21803:(PDF) 21743:S2CID 21723:(PDF) 21638:(PDF) 21520:JSTOR 21454:Notes 21368:+ 001 21332:+ 006 21312:+ 006 21257:+ 532 21230:+ 532 20845:P(x) 20839:+ 001 20803:+ 006 20783:+ 006 19483:+ 001 18854:' 18710:alpha 18476:alpha 18359:break 18155:while 18095:zeros 17825:alpha 17771:alpha 17732:zeros 17678:floor 17447:alpha 17390:alpha 16583:) to 16436:+ 732 16418:+ 001 16200:(0): 16191:) by 16083:/2 16068:while 15078:= 3: 14610:), Λ( 12814:with 12812:(929) 7401:over 7224:is a 7068:, of 6887:with 3959:< 3353:from 3095:over 2624:is a 2182:with 1898:every 1386:is a 1366:< 1264:with 995:with 926:size 783:Years 730:with 689:CCSDS 584:Wuala 485:DVB-S 338:burst 275:WiMAX 159:-code 22482:NASA 22424:ISBN 22401:ISBN 22317:ISBN 22297:OCLC 22233:IT-6 22176:ISBN 22137:2010 21985:ISBN 21938:ISBN 21759:See 21698:OCLC 21688:ISBN 21664:ISBN 21578:OCLC 21568:ISBN 21543:OCLC 21352:) = 21344:) / 21242:) = 21207:) = 21171:001 21160:000 21046:for 20823:) = 20815:) / 19453:− 1 19158:xpad 19152:< 19123:x, k 19113:xpad 19054:find 19008:trim 18524:find 18443:trim 18395:trim 18377:trim 18251:trim 18227:conv 18158:true 18017:trim 17951:Synd 17864:Synd 17858:trim 17807:Synd 17617:conv 17310:and 17308:LDPC 17224:and 17166:The 17134:) − 17126:) = 17097:< 17091:< 16766:Let 16709:Use 16564:) + 16556:) = 16540:) + 16532:) = 16452:) = 16425:) = 16403:) = 16385:608 16378:673 16366:697 16355:683 16347:001 16332:925 16324:000 16305:001 16232:) = 16207:) = 15888:) + 15070:For 14563:173 14552:329 14528:173 14517:634 14490:173 14463:197 14258:The 13018:+ 2 12821:and 12773:and 10210:Let 9003:as: 8993:and 7295:for 7070:data 7053:< 6325:AWGN 1551:< 1096:rate 1006:< 946:, a 786:Code 755:and 685:xDSL 562:CIRC 549:and 496:LDPC 426:and 374:and 287:ATSC 285:and 273:and 251:DVDs 237:and 144:− 1. 44:and 22364:doi 22344:(1) 22264:doi 22237:doi 22206:doi 22160:doi 22084:QEX 22058:doi 21977:doi 21930:doi 21894:doi 21735:doi 21609:doi 21512:doi 21154:−1 20737:003 20730:916 20723:009 20716:007 20709:006 20702:924 20695:006 20563:001 20558:000 20553:000 20548:000 20543:000 20538:000 20533:000 20526:000 20521:001 20516:000 20511:000 20506:000 20501:000 20496:000 20489:000 20484:000 20479:001 20474:000 20469:000 20464:000 20459:000 20452:000 20447:000 20442:000 20437:001 20432:000 20427:000 20422:000 20415:000 20410:000 20405:000 20400:000 20395:001 20390:000 20385:000 20378:000 20373:000 20368:000 20363:000 20358:000 20353:001 20348:000 20341:000 20336:000 20331:000 20326:000 20321:000 20316:000 20311:001 20267:289 20260:637 20253:017 20246:541 20239:437 20232:923 20225:000 20093:562 20088:713 20083:893 20078:923 20073:928 20068:726 20063:121 20056:304 20051:804 20046:904 20041:924 20036:928 20031:430 20026:086 20019:673 20014:865 20009:913 20004:925 19999:928 19994:228 19989:057 19982:848 19977:902 19972:920 19967:926 19962:928 19957:439 19952:456 19945:913 19940:921 19935:925 19930:927 19925:928 19920:246 19915:123 19908:928 19903:928 19898:928 19893:928 19888:928 19883:006 19878:006 19871:000 19866:000 19861:000 19856:000 19851:928 19846:000 19841:001 19348:249 19342:255 19219:to 19170:end 19167:end 19149:len 19128:len 19118:pad 19105:end 19075:end 18995:end 18788:end 18782:err 18770:idx 18746:err 18743:)); 18719:idx 18689:idx 18686:for 18656:(:, 18608:end 18509:)); 18431:end 18365:end 18338:end 18320:all 18290:)); 18269:pad 18215:)); 18194:pad 17963:end 17849:)); 17635:end 17632:end 17581:mod 17572:for 17531:end 17504:g_x 17423:g_x 17191:+ 1 17158:). 16805:to 16572:). 16508:), 16484:), 16302:−1 16250:(0) 16225:(0) 14587:). 14570:412 14567:+ 1 14560:+ 1 14549:576 14546:925 14535:412 14532:+ 1 14525:+ 1 14514:412 14511:762 14500:732 14494:+ 1 14487:846 14484:637 14473:732 14467:+ 1 14460:732 14457:732 14397:). 14216:122 14055:821 14048:329 13986:001 13981:000 13974:000 13969:001 13927:004 13920:167 13898:925 13888:762 13823:762 13818:637 13811:637 13806:732 13779:925 13737:762 13695:637 13641:732 13635:474 13623:487 13604:191 13585:456 13566:123 13465:474 13456:487 13440:191 13424:456 13408:123 13311:474 13302:487 13286:191 13270:382 13140:455 13131:442 13115:738 13099:547 13078:mod 13022:+ 1 12993:522 12984:568 12968:723 12952:809 12826:= 4 12819:= 3 12743:of 12674:). 9835:as 9302:≥ 2 8531:). 8393:of 8232:). 8090:mod 8046:mod 7918:): 7821:): 7724:by 7159:223 7153:255 7112:255 7086:223 7006:255 6797:log 6695:min 6329:FSK 6121:min 6049:mod 5498:mod 5295:by 3123:is 2319:at 1390:of 1156:or 691:'s 551:DVD 547:DAT 360:BCH 326:/2⌋ 283:DVB 271:DSL 247:CDs 109:+ 1 22560:: 22360:11 22358:, 22342:26 22340:, 22336:, 22258:, 22250:; 22231:, 22212:, 22198:, 22156:43 22154:, 22150:, 22082:. 22056:. 22044:49 22042:. 22010:, 21983:. 21975:. 21961:. 21936:. 21928:. 21914:. 21892:, 21880:45 21878:, 21811:51 21809:, 21805:, 21776:. 21741:. 21731:95 21729:. 21725:. 21710:^ 21696:. 21662:, 21658:, 21623:^ 21605:27 21603:. 21599:. 21576:. 21518:. 21506:. 21470:). 21381:) 21187:2 21176:1 21165:0 20293:: 19509:+ 19505:= 19449:= 19251:, 19146:if 19143:); 19102:); 19078:), 19060:gx 19048:gx 19042:gf 19036:gt 19018:gx 19003:gt 18974:); 18926:); 18905:gf 18851:)) 18830:gf 18809:er 18803:gf 18776::) 18764:er 18713:.^ 18683:); 18632:); 18590:); 18554:if 18545:); 18539:), 18533:== 18518:gf 18479:.^ 18452:); 18419:g1 18413:g1 18407:g0 18404:); 18389:r1 18386:); 18383:r1 18371:r0 18350:== 18317:if 18302:g0 18287:g0 18260:); 18242:); 18239:g1 18212:g1 18182:); 18179:r1 18173:r0 18143:f0 18137:g1 18131:f1 18125:g0 18122:); 18110:), 18089:gf 18083:f1 18080:); 18068:(, 18065:gf 18059:f0 18047:r1 18044:); 18041:r0 18026:); 18023:r0 18011:r0 18008:); 17993:r0 17987:gf 17981:r0 17972:r0 17915:); 17873:if 17867:); 17828:.^ 17798:); 17777:gf 17747:); 17702:); 17681:(( 17629:); 17507:); 17480:); 17468:(, 17465:gf 17450:); 17417:); 17396:gf 17207:⌊( 17199:⌊( 16421:Ω( 16399:Λ( 16375:2 16352:1 16329:0 16265:. 16241:/ 16228:Ω( 16216:/ 16203:Λ( 16175:-1 16164:- 16162:-2 16144:-1 16133:- 16131:-2 16113:-1 16105:/ 16103:-2 16079:≥ 16053:−1 16048:) 16028:−1 16003:). 15905:= 15901:) 15880:) 14599:. 14573:2 14538:1 14503:2 14476:1 14423:+1 14270:: 14250:. 14200:74 14107:. 13953:: 13485:. 12810:GF 12708:. 10958:0. 10772:0. 10538:0. 10384:0. 9998:Λ( 9829:Λ( 9821:. 9298:− 9290:. 7677:. 6824:, 6776:, 6718:, 6331:: 6263:≤ 6259:+ 6169:. 6141:1. 5928:: 5763:: 5439:: 4734:. 4125:. 3971:: 3775:A 3767:. 3429:. 3143:. 2732:: 1950:. 1896:, 1394:. 1374:, 1354:, 1292:≤ 1188:. 774:: 767:. 761:dB 751:, 747:, 723:. 713:CC 699:. 606:, 602:, 598:, 525:. 518:. 502:. 441:. 434:. 419:. 318:or 304:= 293:. 277:, 265:, 261:, 253:, 249:, 245:, 140:= 126:≥ 122:= 105:− 22366:: 22266:: 22260:8 22239:: 22208:: 22162:: 22100:. 22064:. 22060:: 22014:: 21993:. 21979:: 21969:: 21946:. 21932:: 21922:: 21896:: 21832:. 21787:. 21749:. 21737:: 21704:. 21617:. 21611:: 21584:. 21549:. 21526:. 21514:: 21508:9 21405:c 21397:b 21390:x 21388:( 21386:E 21379:x 21377:( 21375:P 21366:x 21362:x 21358:x 21356:( 21354:P 21350:x 21348:( 21346:E 21342:x 21340:( 21338:Q 21330:x 21326:x 21322:x 21320:( 21318:E 21310:x 21306:x 21302:x 21298:x 21294:x 21292:( 21290:Q 21283:x 21281:( 21279:E 21275:x 21273:( 21271:E 21267:x 21265:( 21263:Q 21255:x 21251:x 21247:2 21244:A 21240:x 21238:( 21236:E 21228:x 21224:x 21220:x 21216:x 21212:2 21209:R 21205:x 21203:( 21201:Q 21147:i 21144:A 21139:i 21136:R 21132:i 21113:1 21110:= 21105:0 21101:A 21079:0 21076:= 21071:1 21064:A 21052:n 21048:i 21034:} 21031:) 21026:i 21022:a 21018:( 21015:b 21012:, 21007:i 21003:a 20999:{ 20979:= 20974:0 20970:R 20948:) 20943:i 20939:a 20932:x 20929:( 20924:n 20919:1 20916:= 20913:i 20905:= 20900:1 20893:R 20862:c 20854:b 20837:x 20833:x 20829:x 20827:( 20825:P 20821:x 20819:( 20817:E 20813:x 20811:( 20809:Q 20801:x 20797:x 20793:x 20791:( 20789:E 20781:x 20777:x 20773:x 20769:x 20765:x 20763:( 20761:Q 20743:] 20689:[ 20684:= 20679:] 20671:4 20667:q 20657:3 20653:q 20643:2 20639:q 20629:1 20625:q 20615:0 20611:q 20601:1 20597:e 20587:0 20583:e 20576:[ 20569:] 20305:[ 20273:] 20219:[ 20214:= 20209:] 20201:4 20197:q 20187:3 20183:q 20173:2 20169:q 20159:1 20155:q 20145:0 20141:q 20131:1 20127:e 20117:0 20113:e 20106:[ 20099:] 19835:[ 19810:2 19805:i 19801:a 19795:i 19791:b 19784:= 19781:) 19776:4 19771:i 19767:a 19761:4 19757:q 19753:+ 19748:3 19743:i 19739:a 19733:3 19729:q 19725:+ 19720:2 19715:i 19711:a 19705:2 19701:q 19697:+ 19692:i 19688:a 19682:1 19678:q 19674:+ 19669:0 19665:q 19661:( 19655:) 19650:i 19646:a 19640:1 19636:e 19632:+ 19627:0 19623:e 19619:( 19614:i 19610:b 19597:e 19581:0 19578:= 19575:) 19570:i 19566:a 19562:( 19559:Q 19553:) 19548:i 19544:a 19540:( 19537:E 19532:i 19528:b 19511:e 19507:c 19503:b 19492:c 19481:x 19477:x 19473:x 19471:( 19469:p 19458:a 19451:i 19446:i 19442:a 19420:n 19400:) 19395:3 19391:n 19387:( 19384:O 19351:) 19345:, 19339:( 19316:! 19313:k 19310:! 19307:) 19304:k 19298:n 19295:( 19290:! 19287:n 19281:= 19275:) 19270:k 19267:n 19262:( 19232:n 19228:a 19205:1 19201:a 19164:; 19161:= 19155:k 19140:x 19137:( 19131:= 19125:) 19121:( 19115:= 19096:. 19093:g 19090:, 19087:m 19084:. 19081:g 19072:: 19069:) 19066:1 19063:, 19057:( 19051:( 19045:( 19039:= 19033:; 19030:x 19027:. 19024:g 19021:= 19015:) 19013:g 19011:( 19005:= 18992:; 18989:x 18986:. 18980:= 18971:k 18968:: 18965:1 18962:( 18956:+ 18953:) 18950:k 18947:: 18944:1 18941:( 18935:= 18920:, 18917:m 18914:, 18908:( 18902:= 18893:; 18890:x 18887:. 18881:= 18878:) 18875:x 18872:. 18866:( 18857:; 18845:, 18842:m 18839:, 18836:b 18833:( 18827:\ 18824:) 18818:, 18815:m 18812:, 18806:( 18800:( 18797:= 18785:; 18779:= 18773:, 18767:( 18761:; 18758:x 18755:. 18752:e 18749:= 18740:x 18737:. 18731:- 18728:n 18725:( 18722:* 18716:( 18707:= 18704:e 18698:: 18695:1 18692:= 18677:: 18674:1 18671:( 18665:= 18662:) 18659:1 18653:b 18650:; 18647:x 18644:. 18638:= 18626:( 18620:= 18605:; 18599:; 18596:= 18587:k 18584:: 18581:1 18578:( 18572:= 18566:) 18560:( 18542:m 18536:0 18527:( 18521:( 18515:= 18506:0 18503:: 18500:1 18497:- 18494:: 18491:1 18488:- 18485:n 18482:( 18473:, 18470:g 18467:( 18461:= 18449:g 18446:( 18440:= 18437:g 18428:; 18425:g 18422:= 18416:; 18410:= 18398:( 18392:= 18380:( 18374:= 18362:; 18356:) 18353:0 18347:) 18341:- 18335:: 18332:1 18329:( 18323:( 18311:; 18308:c 18305:- 18299:= 18296:g 18284:( 18278:, 18275:c 18272:( 18266:= 18263:c 18257:c 18254:( 18248:= 18245:c 18236:, 18230:( 18224:= 18221:c 18209:( 18203:, 18197:( 18191:= 18176:, 18170:( 18164:= 18146:; 18140:= 18134:; 18128:= 18116:, 18113:m 18104:, 18101:1 18098:( 18092:( 18086:= 18074:, 18071:m 18062:= 18056:; 18050:= 18038:( 18032:= 18020:( 18014:= 18002:, 17999:m 17996:, 17990:( 17984:= 17978:; 17975:= 17960:; 17954:; 17948:= 17945:S 17942:; 17939:= 17936:g 17933:; 17930:= 17924:; 17921:= 17912:k 17909:: 17906:1 17903:( 17897:= 17891:) 17888:x 17885:. 17879:( 17861:( 17855:= 17846:k 17843:- 17840:n 17837:: 17834:1 17831:( 17822:, 17816:( 17810:= 17792:, 17789:m 17786:, 17783:2 17780:( 17774:= 17765:; 17762:= 17759:S 17756:; 17753:= 17750:g 17744:n 17741:, 17738:1 17735:( 17729:= 17720:; 17717:x 17714:. 17708:= 17699:2 17696:/ 17693:) 17690:k 17687:- 17684:n 17675:= 17660:) 17656:( 17626:, 17623:g 17620:( 17614:= 17611:g 17608:) 17605:n 17602:, 17599:k 17596:- 17593:n 17590:: 17587:1 17584:( 17578:= 17575:k 17566:; 17563:1 17560:= 17557:g 17554:) 17550:( 17544:= 17542:g 17528:; 17522:- 17516:= 17501:, 17495:( 17489:= 17474:, 17471:m 17462:= 17444:, 17441:n 17438:, 17435:k 17432:( 17426:= 17411:, 17408:m 17405:, 17402:2 17399:( 17393:= 17363:) 17359:( 17353:= 17290:k 17286:n 17264:) 17259:m 17255:2 17251:( 17248:F 17245:G 17213:k 17209:n 17201:d 17195:d 17189:k 17185:n 17180:k 17178:, 17176:n 17172:d 17156:x 17154:( 17152:c 17148:x 17146:( 17144:C 17140:x 17138:( 17136:E 17132:x 17130:( 17128:R 17124:x 17122:( 17120:C 17100:n 17094:j 17088:t 17078:) 17073:v 17067:j 17063:E 17057:v 17049:+ 17043:+ 17038:2 17032:j 17028:E 17022:2 17014:+ 17009:1 17003:j 16999:E 16993:1 16985:( 16979:= 16970:j 16966:E 16958:) 16953:1 16949:E 16943:1 16937:v 16929:+ 16923:+ 16918:1 16912:v 16908:E 16902:1 16894:+ 16889:v 16885:E 16881:( 16874:v 16866:1 16858:= 16849:0 16845:E 16818:t 16814:E 16791:1 16787:E 16776:x 16774:( 16772:E 16768:v 16749:t 16745:R 16722:1 16718:R 16695:t 16689:j 16683:1 16674:) 16669:j 16661:( 16658:r 16655:= 16650:j 16646:S 16642:= 16637:j 16633:E 16629:= 16624:j 16620:R 16609:x 16607:( 16605:E 16601:x 16599:( 16597:R 16593:t 16589:x 16587:( 16585:R 16581:x 16579:( 16577:r 16570:x 16568:( 16566:E 16562:x 16560:( 16558:C 16554:x 16552:( 16550:R 16546:x 16544:( 16542:e 16538:x 16536:( 16534:c 16530:x 16528:( 16526:r 16522:x 16520:( 16518:r 16514:x 16512:( 16510:e 16506:x 16504:( 16502:c 16498:x 16496:( 16494:R 16490:x 16488:( 16486:E 16482:x 16480:( 16478:C 16474:x 16472:( 16470:e 16466:x 16464:( 16462:r 16458:x 16456:( 16454:s 16450:x 16448:( 16446:c 16434:x 16430:2 16427:R 16423:x 16416:x 16412:x 16408:2 16405:A 16401:x 16391:x 16387:x 16380:x 16368:x 16361:x 16357:x 16342:x 16338:x 16334:x 16319:x 16315:x 16311:x 16307:x 16295:i 16292:A 16287:i 16284:R 16280:i 16263:i 16258:i 16254:A 16247:i 16243:A 16238:i 16234:R 16230:x 16222:i 16218:A 16213:i 16209:A 16205:x 16197:i 16193:A 16189:x 16185:x 16181:x 16173:i 16169:A 16166:Q 16160:i 16156:A 16151:i 16147:A 16142:i 16138:R 16135:Q 16129:i 16125:R 16120:i 16116:R 16111:i 16107:R 16101:i 16097:R 16093:Q 16089:i 16085:i 16081:t 16076:i 16072:R 16064:i 16060:0 16057:A 16050:A 16046:x 16044:( 16042:S 16038:0 16035:R 16032:x 16025:R 16019:x 16017:( 16015:Q 16011:x 16009:( 16007:B 16001:x 15997:x 15995:( 15992:i 15988:R 15984:) 15982:x 15978:x 15976:( 15973:i 15969:B 15965:) 15963:x 15959:x 15957:( 15954:i 15950:A 15944:t 15940:x 15938:( 15935:i 15931:R 15927:i 15923:R 15918:) 15916:x 15914:( 15911:i 15907:R 15903:x 15899:x 15897:( 15894:i 15890:B 15886:x 15884:( 15882:S 15878:x 15876:( 15873:i 15869:A 15846:] 15838:0 15826:x 15821:1 15807:2 15803:x 15797:2 15785:0 15778:0 15771:0 15762:6 15758:x 15752:0 15748:Q 15738:7 15734:x 15728:1 15724:Q 15714:8 15710:x 15704:2 15700:Q 15693:[ 15688:= 15683:] 15675:1 15671:S 15663:x 15656:1 15652:S 15646:1 15638:+ 15633:2 15629:S 15619:2 15615:x 15607:1 15603:S 15597:2 15589:+ 15584:2 15580:S 15574:1 15566:+ 15561:3 15557:S 15547:3 15543:x 15535:1 15531:S 15525:3 15517:+ 15512:2 15508:S 15502:2 15494:+ 15489:3 15485:S 15479:1 15471:+ 15466:4 15462:S 15452:4 15448:x 15440:2 15436:S 15430:3 15422:+ 15417:3 15413:S 15407:2 15399:+ 15394:4 15390:S 15384:1 15376:+ 15371:5 15367:S 15357:5 15353:x 15345:3 15341:S 15335:3 15327:+ 15322:4 15318:S 15312:2 15304:+ 15299:5 15295:S 15289:1 15281:+ 15276:6 15272:S 15262:6 15258:x 15250:4 15246:S 15240:3 15232:+ 15227:5 15223:S 15217:2 15209:+ 15204:6 15200:S 15194:1 15180:7 15176:x 15168:5 15164:S 15158:3 15150:+ 15145:6 15141:S 15135:2 15121:8 15117:x 15109:6 15105:S 15099:3 15088:[ 15076:e 15072:t 15056:) 15053:x 15050:( 15044:+ 15039:t 15035:x 15031:) 15028:x 15025:( 15022:Q 15019:= 15016:) 15013:x 15010:( 15007:S 15004:) 15001:x 14998:( 14967:0 14959:+ 14956:x 14951:1 14943:+ 14937:+ 14932:1 14926:e 14922:x 14916:1 14910:e 14902:+ 14897:e 14893:x 14887:e 14879:= 14872:) 14869:x 14866:( 14856:1 14853:+ 14850:x 14845:1 14837:+ 14831:+ 14826:1 14820:e 14816:x 14810:1 14804:e 14796:+ 14791:e 14787:x 14781:e 14773:= 14766:) 14763:x 14760:( 14748:1 14744:S 14740:+ 14737:x 14732:2 14728:S 14724:+ 14718:+ 14713:2 14707:t 14703:x 14697:1 14691:t 14687:S 14683:+ 14678:1 14672:t 14668:x 14662:t 14658:S 14654:= 14647:) 14644:x 14641:( 14638:S 14624:e 14620:t 14616:x 14612:x 14608:x 14606:( 14604:S 14585:x 14581:C 14565:x 14558:x 14554:x 14543:3 14530:x 14523:x 14519:x 14508:2 14497:1 14492:x 14481:1 14470:1 14465:x 14454:0 14448:m 14443:b 14438:B 14433:C 14428:d 14421:n 14417:S 14412:n 14395:x 14391:x 14389:( 14387:C 14379:e 14375:x 14359:e 14353:i 14349:S 14340:e 14332:+ 14326:+ 14321:1 14315:i 14311:S 14302:1 14294:+ 14289:i 14285:S 14281:= 14268:e 14264:x 14248:s 14244:x 14242:( 14240:r 14224:4 14220:x 14213:+ 14208:3 14204:x 14197:= 14192:4 14188:x 14182:2 14178:e 14174:+ 14169:3 14165:x 14159:1 14155:e 14141:2 14137:2 14133:2 14131:e 14126:1 14122:1 14118:1 14116:e 14089:2 14086:x 14082:1 14079:x 14061:] 14042:[ 14037:= 14032:] 14024:1 14010:2 13999:[ 13992:] 13963:[ 13933:] 13914:[ 13909:= 13904:] 13879:[ 13874:= 13869:] 13861:1 13847:2 13836:[ 13829:] 13800:[ 13776:= 13773:) 13768:4 13764:3 13760:( 13757:r 13754:= 13749:4 13745:S 13740:, 13734:= 13731:) 13726:3 13722:3 13718:( 13715:r 13712:= 13707:3 13703:S 13698:, 13692:= 13689:) 13684:2 13680:3 13676:( 13673:r 13670:= 13665:2 13661:S 13638:= 13632:+ 13629:3 13620:+ 13615:2 13611:3 13601:+ 13596:3 13592:3 13582:+ 13577:4 13573:3 13563:+ 13558:5 13554:3 13547:2 13544:+ 13539:6 13535:3 13528:3 13525:= 13522:) 13517:1 13513:3 13509:( 13506:r 13503:= 13498:1 13494:S 13483:α 13479:r 13462:+ 13459:x 13453:+ 13448:2 13444:x 13437:+ 13432:3 13428:x 13421:+ 13416:4 13412:x 13405:+ 13400:5 13396:x 13392:2 13389:+ 13384:6 13380:x 13376:3 13373:= 13370:) 13367:x 13364:( 13361:e 13358:+ 13355:) 13352:x 13349:( 13346:s 13343:= 13340:) 13337:x 13334:( 13331:r 13308:+ 13305:x 13299:+ 13294:2 13290:x 13283:+ 13278:3 13274:x 13267:+ 13262:4 13258:x 13254:1 13251:+ 13246:5 13242:x 13238:2 13235:+ 13230:6 13226:x 13222:3 13219:= 13216:) 13213:x 13210:( 13205:r 13201:s 13192:t 13188:x 13183:) 13180:x 13177:( 13174:p 13171:= 13168:) 13165:x 13162:( 13159:s 13137:+ 13134:x 13128:+ 13123:2 13119:x 13112:+ 13107:3 13103:x 13096:= 13093:) 13090:x 13087:( 13082:g 13072:t 13068:x 13063:) 13060:x 13057:( 13054:p 13051:= 13048:) 13045:x 13042:( 13037:r 13033:s 13020:x 13016:x 13012:x 13010:( 13008:p 12990:+ 12987:x 12981:+ 12976:2 12972:x 12965:+ 12960:3 12956:x 12949:+ 12944:4 12940:x 12936:= 12933:) 12928:4 12924:3 12917:x 12914:( 12911:) 12906:3 12902:3 12895:x 12892:( 12889:) 12884:2 12880:3 12873:x 12870:( 12867:) 12864:3 12858:x 12855:( 12852:= 12849:) 12846:x 12843:( 12840:g 12824:t 12817:α 12797:x 12795:( 12793:s 12789:x 12787:( 12785:r 12779:k 12777:i 12775:e 12770:k 12768:i 12764:x 12762:( 12760:e 12747:k 12745:X 12720:k 12718:i 12697:k 12695:Y 12690:k 12688:X 12660:1 12652:k 12648:X 12625:k 12621:X 12609:k 12607:X 12602:i 12585:ν 12580:ν 12576:ν 12572:ν 12556:] 12545:+ 12538:S 12518:2 12515:+ 12508:S 12495:1 12492:+ 12485:S 12475:[ 12470:= 12465:] 12457:1 12436:1 12405:[ 12398:] 12390:1 12381:2 12377:S 12364:1 12361:+ 12354:S 12342:S 12310:1 12307:+ 12300:S 12287:3 12283:S 12275:2 12271:S 12257:S 12244:2 12240:S 12232:1 12228:S 12221:[ 12208:i 12203:v 12199:v 12195:j 12174:+ 12171:j 12167:S 12160:= 12155:1 12145:1 12136:+ 12133:j 12129:S 12125:+ 12119:+ 12114:1 12098:1 12095:+ 12092:j 12088:S 12084:+ 12069:j 12065:S 12039:+ 12036:j 12032:S 12009:0 12006:= 12001:j 11997:S 11983:+ 11978:1 11975:+ 11972:j 11968:S 11962:1 11948:+ 11942:+ 11937:1 11928:+ 11925:j 11921:S 11915:1 11907:+ 11899:+ 11896:j 11892:S 11869:0 11866:= 11862:) 11856:j 11851:k 11847:X 11841:k 11837:Y 11826:1 11823:= 11820:k 11811:( 11797:+ 11791:+ 11787:) 11781:2 11772:+ 11769:j 11764:k 11760:X 11754:k 11750:Y 11739:1 11736:= 11733:k 11724:( 11718:2 11710:+ 11706:) 11700:1 11691:+ 11688:j 11683:k 11679:X 11673:k 11669:Y 11658:1 11655:= 11652:k 11643:( 11637:1 11629:+ 11625:) 11616:+ 11613:j 11608:k 11604:X 11598:k 11594:Y 11583:1 11580:= 11577:k 11568:( 11525:0 11522:= 11518:) 11512:j 11507:k 11503:X 11497:k 11493:Y 11472:1 11469:= 11466:k 11457:( 11453:+ 11447:+ 11443:) 11437:2 11428:+ 11425:j 11420:k 11416:X 11410:k 11406:Y 11400:2 11385:1 11382:= 11379:k 11370:( 11366:+ 11362:) 11356:1 11347:+ 11344:j 11339:k 11335:X 11329:k 11325:Y 11319:1 11304:1 11301:= 11298:k 11289:( 11285:+ 11281:) 11272:+ 11269:j 11264:k 11260:X 11254:k 11250:Y 11239:1 11236:= 11233:k 11224:( 11201:0 11198:= 11194:) 11188:j 11183:k 11179:X 11173:k 11169:Y 11155:+ 11149:+ 11144:2 11135:+ 11132:j 11127:k 11123:X 11117:k 11113:Y 11107:2 11099:+ 11094:1 11085:+ 11082:j 11077:k 11073:X 11067:k 11063:Y 11057:1 11049:+ 11041:+ 11038:j 11033:k 11029:X 11023:k 11019:Y 11014:( 11003:1 11000:= 10997:k 10982:ν 10978:k 10955:= 10950:j 10945:k 10941:X 10935:k 10931:Y 10917:+ 10911:+ 10906:2 10897:+ 10894:j 10889:k 10885:X 10879:k 10875:Y 10869:2 10861:+ 10856:1 10847:+ 10844:j 10839:k 10835:X 10829:k 10825:Y 10819:1 10811:+ 10803:+ 10800:j 10795:k 10791:X 10785:k 10781:Y 10769:= 10756:k 10752:X 10743:+ 10740:j 10735:k 10731:X 10725:k 10721:Y 10707:+ 10701:+ 10696:2 10688:k 10684:X 10675:+ 10672:j 10667:k 10663:X 10657:k 10653:Y 10647:2 10639:+ 10634:1 10626:k 10622:X 10613:+ 10610:j 10605:k 10601:X 10595:k 10591:Y 10585:1 10577:+ 10569:+ 10566:j 10561:k 10557:X 10551:k 10547:Y 10535:= 10531:) 10517:k 10513:X 10499:+ 10493:+ 10488:2 10480:k 10476:X 10470:2 10462:+ 10457:1 10449:k 10445:X 10439:1 10431:+ 10428:1 10424:( 10415:+ 10412:j 10407:k 10403:X 10397:k 10393:Y 10381:= 10378:) 10373:1 10365:k 10361:X 10357:( 10346:+ 10343:j 10338:k 10334:X 10328:k 10324:Y 10293:+ 10290:j 10285:k 10281:X 10275:k 10271:Y 10244:j 10238:1 10218:j 10196:0 10193:= 10190:) 10185:1 10177:k 10173:X 10169:( 10146:0 10143:= 10140:1 10134:1 10131:= 10128:) 10123:k 10119:X 10110:1 10102:k 10098:X 10091:1 10088:( 10066:1 10058:k 10054:X 10050:= 10047:x 10025:1 10017:k 10013:X 10002:) 10000:x 9976:x 9962:+ 9956:+ 9951:2 9947:x 9941:2 9933:+ 9928:1 9924:x 9918:1 9910:+ 9907:1 9904:= 9901:) 9896:k 9892:X 9888:x 9882:1 9879:( 9869:1 9866:= 9863:k 9855:= 9852:) 9849:x 9846:( 9833:) 9831:x 9818:k 9816:X 9794:v 9774:v 9771:2 9748:k 9742:n 9731:k 9729:X 9718:k 9712:k 9710:X 9692:] 9684:k 9678:n 9674:S 9657:2 9653:S 9643:1 9639:S 9632:[ 9627:= 9622:] 9610:Y 9593:2 9589:Y 9579:1 9575:Y 9568:[ 9561:] 9553:k 9547:n 9538:X 9525:k 9519:n 9514:2 9510:X 9502:k 9496:n 9491:1 9487:X 9455:2 9446:X 9433:2 9428:2 9424:X 9416:2 9411:1 9407:X 9397:1 9388:X 9375:1 9370:2 9366:X 9358:1 9353:1 9349:X 9342:[ 9329:k 9327:Y 9322:k 9320:X 9315:k 9313:X 9309:ν 9304:ν 9300:k 9296:n 9276:j 9271:k 9267:X 9263:= 9258:j 9254:) 9246:k 9242:i 9233:( 9230:= 9223:k 9219:i 9212:j 9204:= 9197:k 9193:i 9188:) 9182:j 9174:( 9150:j 9145:k 9141:X 9135:k 9131:Y 9120:1 9117:= 9114:k 9106:= 9101:j 9097:S 9070:k 9066:i 9061:e 9057:= 9052:k 9048:Y 9041:, 9034:k 9030:i 9021:= 9016:k 9012:X 9000:k 8998:Y 8990:k 8988:X 8957:j 8932:) 8929:x 8926:( 8923:s 8903:0 8900:= 8897:) 8892:j 8884:( 8881:s 8857:k 8851:n 8848:, 8842:, 8839:2 8836:, 8833:1 8830:= 8827:j 8823:, 8816:k 8812:i 8806:) 8801:j 8793:( 8784:k 8780:i 8775:e 8764:1 8761:= 8758:k 8750:= 8740:) 8735:j 8727:( 8724:e 8721:= 8711:) 8706:j 8698:( 8695:e 8692:+ 8689:0 8686:= 8683:) 8678:j 8670:( 8667:e 8664:+ 8661:) 8656:j 8648:( 8645:s 8642:= 8639:) 8634:j 8626:( 8623:r 8620:= 8611:j 8607:S 8591:j 8587:S 8571:k 8565:n 8552:1 8529:x 8527:( 8525:s 8521:x 8519:( 8517:r 8513:x 8511:( 8509:e 8503:k 8501:i 8499:e 8494:k 8492:i 8488:ν 8468:k 8464:i 8459:x 8451:k 8447:i 8442:e 8431:1 8428:= 8425:k 8417:= 8414:) 8411:x 8408:( 8405:e 8395:x 8390:k 8388:i 8384:ν 8380:x 8375:i 8373:e 8355:i 8351:x 8345:i 8341:e 8335:1 8329:n 8324:0 8321:= 8318:i 8310:= 8307:) 8304:x 8301:( 8298:e 8279:) 8276:x 8273:( 8270:e 8267:+ 8264:) 8261:x 8258:( 8255:s 8252:= 8249:) 8246:x 8243:( 8240:r 8230:x 8228:( 8226:r 8222:x 8220:( 8218:e 8202:k 8196:n 8193:, 8187:, 8184:2 8181:, 8178:1 8175:= 8172:j 8166:, 8163:0 8160:= 8157:) 8152:j 8144:( 8141:s 8121:0 8118:= 8115:) 8110:j 8099:x 8094:( 8086:) 8083:x 8080:( 8077:g 8074:= 8071:) 8066:j 8055:x 8050:( 8042:) 8039:x 8036:( 8033:s 8023:x 8021:( 8019:g 8015:x 8013:( 8011:s 8004:α 7990:, 7987:) 7982:j 7971:x 7968:( 7963:k 7957:n 7952:1 7949:= 7946:j 7938:= 7935:) 7932:x 7929:( 7926:g 7916:x 7914:( 7912:g 7908:x 7906:( 7904:s 7886:i 7882:x 7876:i 7872:c 7866:1 7860:n 7855:0 7852:= 7849:i 7841:= 7838:) 7835:x 7832:( 7829:s 7819:x 7817:( 7815:s 7801:) 7796:1 7790:n 7786:c 7782:, 7776:, 7771:i 7767:c 7763:, 7757:, 7752:0 7748:c 7744:( 7657:p 7631:1 7623:= 7618:q 7593:) 7590:) 7585:q 7577:( 7574:p 7571:, 7565:, 7562:) 7557:2 7549:( 7546:p 7543:( 7523:) 7520:a 7514:( 7511:p 7505:a 7485:) 7482:) 7477:1 7471:q 7463:( 7460:p 7457:, 7451:, 7448:) 7443:1 7435:( 7432:p 7429:( 7409:F 7389:p 7368:| 7364:F 7360:| 7356:= 7353:q 7333:} 7330:1 7324:q 7321:, 7315:, 7312:1 7309:{ 7303:i 7281:i 7256:F 7236:F 7162:) 7156:, 7150:( 7147:= 7144:) 7141:k 7138:, 7135:n 7132:( 7109:= 7106:n 7083:= 7080:k 7056:n 7050:k 7030:k 7003:= 7000:1 6992:8 6988:2 6984:= 6981:n 6961:1 6953:m 6949:2 6945:= 6942:n 6922:m 6900:m 6896:2 6875:F 6852:M 6832:s 6809:M 6801:2 6792:m 6787:= 6784:h 6762:h 6758:) 6754:s 6748:1 6745:( 6739:1 6736:= 6731:s 6727:P 6706:) 6703:1 6691:d 6687:( 6682:2 6679:1 6674:= 6671:t 6643:n 6639:) 6633:s 6629:P 6622:1 6619:( 6609:s 6605:P 6598:) 6590:n 6585:( 6574:n 6569:1 6566:+ 6563:t 6560:= 6547:n 6544:1 6537:m 6534:1 6524:b 6520:P 6491:n 6487:) 6481:s 6477:P 6470:1 6467:( 6457:s 6453:P 6446:) 6438:n 6433:( 6422:n 6417:1 6414:+ 6411:t 6408:= 6395:n 6392:1 6384:1 6376:m 6372:2 6365:1 6359:m 6355:2 6344:b 6340:P 6300:S 6280:E 6269:k 6265:n 6261:S 6257:E 6255:2 6226:2 6222:/ 6218:) 6215:k 6209:n 6206:( 6186:k 6180:n 6159:k 6155:n 6138:+ 6135:k 6129:n 6126:= 6117:d 6103:k 6096:F 6092:n 6067:. 6063:) 6060:x 6057:( 6054:g 6045:0 6039:) 6036:x 6033:( 6028:r 6024:s 6017:) 6014:x 6011:( 6006:r 6002:s 5995:) 5992:x 5989:( 5984:r 5980:s 5971:t 5967:x 5960:) 5957:x 5954:( 5951:p 5945:) 5942:x 5939:( 5936:s 5916:) 5913:x 5910:( 5907:g 5886:C 5865:) 5862:x 5859:( 5856:s 5834:. 5830:) 5827:x 5824:( 5819:r 5815:s 5806:t 5802:x 5795:) 5792:x 5789:( 5786:p 5783:= 5780:) 5777:x 5774:( 5771:s 5751:) 5748:x 5745:( 5742:p 5722:k 5702:) 5699:x 5696:( 5693:s 5671:t 5667:x 5660:) 5657:x 5654:( 5651:p 5629:0 5625:x 5621:, 5616:1 5612:x 5608:, 5602:, 5597:2 5591:t 5587:x 5583:, 5578:1 5572:t 5568:x 5547:1 5541:t 5519:. 5516:) 5513:x 5510:( 5507:g 5489:t 5485:x 5478:) 5475:x 5472:( 5469:p 5466:= 5463:) 5460:x 5457:( 5452:r 5448:s 5427:) 5424:x 5421:( 5416:r 5412:s 5391:t 5371:) 5368:x 5365:( 5362:g 5342:k 5336:n 5333:= 5330:t 5308:t 5304:x 5283:) 5280:x 5277:( 5274:p 5251:) 5248:x 5245:( 5242:p 5222:) 5219:x 5216:( 5213:s 5193:) 5190:x 5187:( 5184:p 5164:) 5161:x 5158:( 5155:g 5135:) 5132:x 5129:( 5126:s 5106:) 5103:x 5100:( 5097:s 5077:) 5074:x 5071:( 5068:p 5048:k 5028:) 5025:x 5022:( 5019:s 4999:) 4996:x 4993:( 4990:g 4987:) 4984:x 4981:( 4978:p 4975:= 4972:) 4969:x 4966:( 4963:s 4932:. 4928:} 4922:k 4916:n 4908:, 4902:, 4897:2 4889:, 4884:1 4869:i 4865:a 4859:i 4855:s 4849:n 4844:1 4841:= 4838:i 4830:= 4827:) 4824:a 4821:( 4818:s 4812:| 4805:) 4799:n 4795:s 4791:, 4785:, 4780:2 4776:s 4772:, 4767:1 4763:s 4758:( 4753:{ 4749:= 4745:C 4722:1 4719:= 4716:i 4692:k 4686:n 4682:x 4678:+ 4673:1 4667:k 4661:n 4657:x 4651:1 4645:k 4639:n 4635:g 4631:+ 4625:+ 4622:x 4617:1 4613:g 4609:+ 4604:0 4600:g 4596:= 4592:) 4586:1 4580:k 4574:n 4571:+ 4568:i 4557:x 4553:( 4545:) 4539:1 4536:+ 4533:i 4522:x 4518:( 4513:) 4507:i 4496:x 4492:( 4488:= 4485:) 4482:x 4479:( 4476:g 4437:) 4434:x 4431:( 4428:g 4408:k 4402:n 4382:) 4379:x 4376:( 4373:g 4350:1 4344:k 4324:) 4321:x 4318:( 4315:p 4295:) 4292:x 4289:( 4286:s 4266:) 4263:x 4260:( 4257:s 4237:1 4231:q 4225:n 4205:1 4199:n 4179:) 4176:x 4173:( 4170:s 4150:) 4147:x 4144:( 4141:p 4106:} 4101:1 4095:n 4087:, 4081:, 4076:2 4068:, 4062:, 4059:1 4056:{ 4053:= 4048:1 4042:n 4038:a 4034:, 4028:, 4023:0 4019:a 3997:i 3989:= 3984:i 3980:a 3969:α 3965:k 3961:q 3957:n 3939:] 3933:) 3928:1 3922:n 3918:a 3914:( 3909:m 3905:p 3890:) 3885:1 3881:a 3877:( 3872:m 3868:p 3860:) 3855:0 3851:a 3847:( 3842:m 3838:p 3831:[ 3826:= 3823:) 3820:m 3817:( 3814:C 3792:m 3788:p 3753:m 3749:p 3726:1 3720:k 3716:m 3712:, 3706:, 3701:0 3697:m 3676:) 3671:1 3665:k 3661:a 3657:( 3652:m 3648:p 3644:, 3638:, 3635:) 3630:0 3626:a 3622:( 3617:m 3613:p 3592:k 3564:] 3558:) 3553:1 3547:n 3543:a 3539:( 3534:m 3530:p 3515:) 3510:1 3506:a 3502:( 3497:m 3493:p 3485:) 3480:0 3476:a 3472:( 3467:m 3463:p 3456:[ 3451:= 3448:) 3445:m 3442:( 3439:C 3415:1 3409:n 3405:a 3401:, 3395:, 3390:k 3386:a 3361:m 3339:m 3335:p 3314:. 3311:} 3308:1 3302:k 3299:, 3293:, 3290:0 3287:{ 3281:i 3271:i 3267:m 3263:= 3260:) 3255:i 3251:a 3247:( 3242:m 3238:p 3217:k 3195:m 3191:p 3168:m 3164:p 3131:A 3103:F 3075:] 3067:1 3061:k 3057:m 3040:1 3036:m 3026:0 3022:m 3015:[ 3008:] 3000:1 2994:k 2989:1 2983:n 2979:a 2966:2 2961:1 2955:n 2951:a 2943:1 2937:n 2933:a 2927:1 2891:1 2885:k 2880:1 2876:a 2863:2 2858:1 2854:a 2846:1 2842:a 2836:1 2827:1 2821:k 2816:0 2812:a 2799:2 2794:0 2790:a 2782:0 2778:a 2772:1 2766:[ 2761:= 2758:m 2755:A 2752:= 2749:) 2746:m 2743:( 2740:C 2720:F 2700:A 2680:k 2674:n 2654:m 2651:A 2648:= 2645:) 2642:m 2639:( 2636:C 2612:C 2590:] 2584:) 2579:1 2573:n 2569:a 2565:( 2560:m 2556:p 2541:) 2536:1 2532:a 2528:( 2523:m 2519:p 2511:) 2506:0 2502:a 2498:( 2493:m 2489:p 2482:[ 2477:= 2474:) 2471:m 2468:( 2465:C 2443:n 2439:F 2430:k 2426:F 2422:: 2419:C 2399:F 2377:1 2371:n 2367:a 2363:, 2357:, 2352:0 2348:a 2327:n 2305:m 2301:p 2280:m 2260:. 2254:i 2250:a 2244:i 2240:m 2234:1 2228:k 2223:0 2220:= 2217:i 2209:= 2206:) 2203:a 2200:( 2195:m 2191:p 2168:m 2164:p 2141:k 2137:F 2130:) 2125:1 2119:k 2115:m 2111:, 2105:, 2100:0 2096:m 2092:( 2089:= 2086:m 2060:k 2056:p 2040:k 2036:a 2032:, 2026:, 2021:1 2017:a 2006:k 1998:p 1990:x 1970:k 1966:q 1955:k 1934:n 1930:/ 1926:1 1923:+ 1920:1 1914:R 1911:+ 1880:n 1876:/ 1872:k 1869:= 1866:R 1846:R 1840:1 1834:n 1830:/ 1826:1 1823:+ 1820:R 1814:1 1811:= 1808:n 1804:/ 1800:1 1797:+ 1794:n 1790:/ 1786:k 1780:1 1777:= 1774:n 1770:/ 1766:d 1763:= 1740:1 1737:+ 1734:k 1728:n 1725:= 1722:d 1698:1 1692:k 1672:1 1669:+ 1666:k 1660:n 1657:= 1654:) 1651:1 1645:k 1642:( 1636:n 1616:1 1610:k 1590:k 1566:. 1560:} 1554:k 1543:F 1535:p 1529:| 1521:) 1516:) 1511:n 1507:a 1503:( 1500:p 1497:, 1491:, 1488:) 1483:2 1479:a 1475:( 1472:p 1469:, 1466:) 1461:1 1457:a 1453:( 1450:p 1445:( 1437:{ 1432:= 1428:C 1406:C 1392:F 1384:α 1380:α 1376:α 1372:α 1368:q 1364:n 1360:α 1356:α 1352:α 1348:n 1344:F 1328:n 1324:a 1320:, 1314:, 1309:1 1305:a 1294:q 1290:n 1286:p 1272:q 1252:F 1242:k 1228:p 1208:k 1176:1 1170:q 1167:= 1164:n 1144:q 1141:= 1138:n 1116:n 1113:k 1108:= 1105:R 1078:q 1058:q 1038:F 1015:q 1009:n 1003:k 993:, 980:k 957:n 934:q 678:K 674:N 670:N 666:K 659:K 655:N 651:N 647:K 643:N 392:n 388:k 353:t 349:b 343:b 334:t 324:t 322:⌊ 314:t 310:k 306:n 302:t 215:e 208:t 201:v 156:q 142:q 138:n 132:p 128:n 124:p 120:q 107:k 103:n 91:k 79:n 20:)

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Knowledge

Reed–Solomon error correction

Index