645:, after him). Parity has a distance of 2, so one bit flip can be detected but not corrected, and any two bit flips will be invisible. The (3,1) repetition has a distance of 3, as three bits need to be flipped in the same triple to obtain another code word with no visible errors. It can correct one-bit errors or it can detect - but not correct - two-bit errors. A (4,1) repetition (each bit is repeated four times) has a distance of 4, so flipping three bits can be detected, but not corrected. When three bits flip in the same group there can be situations where attempting to correct will produce the wrong code word. In general, a code with distance
25:
3361:
490:. If an odd number of bits is changed in transmission, the message will change parity and the error can be detected at this point; however, the bit that changed may have been the parity bit itself. The most common convention is that a parity value of one indicates that there is an odd number of ones in the data, and a parity value of zero indicates that there is an even number of ones. If the number of bits changed is even, the check bit will be valid and the error will not be detected.
3018:
3382:
1918:
91:
3356:{\displaystyle {\vec {x}}={\vec {a}}G={\begin{pmatrix}1&0&1&1\end{pmatrix}}{\begin{pmatrix}1&0&0&0&1&1&0\\0&1&0&0&1&0&1\\0&0&1&0&0&1&1\\0&0&0&1&1&1&1\\\end{pmatrix}}={\begin{pmatrix}1&0&1&1&2&3&2\end{pmatrix}}={\begin{pmatrix}1&0&1&1&0&1&0\end{pmatrix}}}
4297:
4069:
1353:
1346:
1339:
1332:
1325:
1273:
1266:
1259:
1252:
1245:
1238:
1231:
1224:
1198:
1183:
1176:
1169:
1162:
1147:
1140:
1133:
1126:
1106:
1099:
1088:
1081:
1070:
1063:
1052:
1045:
1034:
1027:
1011:
1002:
993:
984:
975:
966:
957:
948:
939:
930:
3839:
3612:
417:). In this context, an extended Hamming code having one extra parity bit is often used. Extended Hamming codes achieve a Hamming distance of four, which allows the decoder to distinguish between when at most one one-bit error occurs and when any two-bit errors occur. In this sense, extended Hamming codes are single-error correcting and double-error detecting, abbreviated as
1368:, identifies the bit in error. If all parity bits are correct, there is no error. Otherwise, the sum of the positions of the erroneous parity bits identifies the erroneous bit. For example, if the parity bits in positions 1, 2 and 8 indicate an error, then bit 1+2+8=11 is in error. If only one parity bit indicates an error, the parity bit itself is in error.
2485:
4081:
3853:
448:, seven-eighths of an inch wide, which had up to six holes per row. During weekdays, when errors in the relays were detected, the machine would stop and flash lights so that the operators could correct the problem. During after-hours periods and on weekends, when there were no operators, the machine simply moved on to the next job.
1891:). Server computers in 21st century, while typically keeping the SECDED level of protection, no longer use the Hamming's method, relying instead on the designs with longer codewords (128 to 256 bits of data) and modified balanced parity-check trees. The (72,64) Hamming code is still popular in some hardware designs, including
2653:
3623:
3399:
659:
Hamming was interested in two problems at once: increasing the distance as much as possible, while at the same time increasing the code rate as much as possible. During the 1940s he developed several encoding schemes that were dramatic improvements on existing codes. The key to all of his systems was
451:
Hamming worked on weekends, and grew increasingly frustrated with having to restart his programs from scratch due to detected errors. In a taped interview, Hamming said, "And so I said, 'Damn it, if the machine can detect an error, why can't it locate the position of the error and correct it?'". Over
1871:
To remedy this shortcoming, Hamming codes can be extended by an extra parity bit. This way, it is possible to increase the minimum distance of the
Hamming code to 4, which allows the decoder to distinguish between single bit errors and two-bit errors. Thus the decoder can detect and correct a single
599:
Such codes cannot correctly repair all errors, however. In our example, if the channel flips two bits and the receiver gets 001, the system will detect the error, but conclude that the original bit is 0, which is incorrect. If we increase the size of the bit string to four, we can detect all two-bit
493:
Moreover, parity does not indicate which bit contained the error, even when it can detect it. The data must be discarded entirely and re-transmitted from scratch. On a noisy transmission medium, a successful transmission could take a long time or may never occur. However, while the quality of parity
1363:
Shown are only 20 encoded bits (5 parity, 15 data) but the pattern continues indefinitely. The key thing about
Hamming Codes that can be seen from visual inspection is that any given bit is included in a unique set of parity bits. To check for errors, check all of the parity bits. The pattern of
412:
Due to the limited redundancy that
Hamming codes add to the data, they can only detect and correct errors when the error rate is low. This is the case in computer memory (usually RAM), where bit errors are extremely rare and Hamming codes are widely used, and a RAM with this correction system is an
580:
will send 111. If the three bits received are not identical, an error occurred during transmission. If the channel is clean enough, most of the time only one bit will change in each triple. Therefore, 001, 010, and 100 each correspond to a 0 bit, while 110, 101, and 011 correspond to a 1 bit, with
672:
of all the bit positions containing a 1) is 0. We use positions 1, 10, 100, etc. (in binary) as the error-correcting bits, which guarantees it is possible to set the error-correcting bits so that the index-XOR of the whole message is 0. If the receiver receives a string with index-XOR 0, they can
612:
If more error-correcting bits are included with a message, and if those bits can be arranged such that different incorrect bits produce different error results, then bad bits could be identified. In a seven-bit message, there are seven possible single bit errors, so three error control bits could
1867:
Hamming codes have a minimum distance of 3, which means that the decoder can detect and correct a single error, but it cannot distinguish a double bit error of some codeword from a single bit error of a different codeword. Thus, some double-bit errors will be incorrectly decoded as if they were
1875:
If the decoder does not attempt to correct errors, it can reliably detect triple bit errors. If the decoder does correct errors, some triple errors will be mistaken for single errors and "corrected" to the wrong value. Error correction is therefore a trade-off between certainty (the ability to
4334:
To decode the
Hamming code, first check the parity bit. If the parity bit indicates an error, single error correction (the Hamming code) will indicate the error location, with "no error" indicating the parity bit. If the parity bit is correct, then single error correction will indicate the
2292:
603:
Moreover, increasing the size of the parity bit string is inefficient, reducing throughput by three times in our original case, and the efficiency drops drastically as we increase the number of times each bit is duplicated in order to detect and correct more errors.
4292:{\displaystyle \mathbf {G} =\left({\begin{array}{cccc|cccc}1&0&0&0&0&1&1&1\\0&1&0&0&1&0&1&1\\0&0&1&0&1&1&0&1\\0&0&0&1&1&1&1&0\end{array}}\right)_{4,8}.}
4064:{\displaystyle \mathbf {H} =\left({\begin{array}{cccc|cccc}0&1&1&1&1&0&0&0\\1&0&1&1&0&1&0&0\\1&1&0&1&0&0&1&0\\1&1&1&0&0&0&0&1\end{array}}\right)_{4,8}.}
553:
possible combinations, enough to represent the digits 0β9. This scheme can detect all single bit-errors, all odd numbered bit-errors and some even numbered bit-errors (for example the flipping of both 1-bits). However it still cannot correct any of these errors.
1931:
In 1950, Hamming introduced the
Hamming code. It encodes four data bits into seven bits by adding three parity bits. As explained earlier, it can either detect and correct single-bit errors or it can detect (but not correct) both single and double-bit errors.
2015:
2494:
2096:
3834:{\displaystyle \mathbf {H} :={\begin{pmatrix}1&0&1&0&1&0&1&0\\0&1&1&0&0&1&1&0\\0&0&0&1&1&1&1&0\\1&1&1&1&1&1&1&1\end{pmatrix}}_{4,8}.}
3607:{\displaystyle \mathbf {G} :={\begin{pmatrix}1&1&1&0&0&0&0&1\\1&0&0&1&1&0&0&1\\0&1&0&1&0&1&0&1\\1&1&0&1&0&0&1&0\end{pmatrix}}_{4,8}}
4074:
For example, the first row in this matrix is the sum of the second and third rows of H in non-systematic form. Using the systematic construction for
Hamming codes from above, the matrix A is apparent and the systematic form of G is written as
452:
the next few years, he worked on the problem of error-correction, developing an increasingly powerful array of algorithms. In 1950, he published what is now known as
Hamming code, which remains in use today in applications such as
2480:{\displaystyle \mathbf {G} :={\begin{pmatrix}1&0&0&0&1&1&0\\0&1&0&0&1&0&1\\0&0&1&0&0&1&1\\0&0&0&1&1&1&1\end{pmatrix}}_{4,7}}
4335:(bitwise) exclusive-or of two error locations. If the locations are equal ("no error") then a double bit error either has not occurred, or has cancelled itself out. Otherwise, a double bit error has occurred.
2164:
581:
the greater quantity of digits that are the same ('0' or a '1') indicating what the data bit should be. A code with this ability to reconstruct the original message in the presence of errors is known as an
383:, which is the highest possible for codes with minimum distance of three (i.e., the minimal number of bit changes needed to go from any code word to any other code word is three) and block length
4328:
where blue digits are data; red digits are parity bits from the
Hamming code; and the green digit is the parity bit added by the code. The green digit makes the parity of the codewords even.
2824:
4425:
1946:
668:
The following general algorithm generates a single-error correcting (SEC) code for any number of bits. The main idea is to choose the error-correcting bits such that the index-XOR (the
4633:
2648:{\displaystyle \mathbf {H} :={\begin{pmatrix}1&1&0&1&1&0&0\\1&0&1&1&0&1&0\\0&1&1&1&0&0&1\end{pmatrix}}_{3,7}.}
2929:
1935:
With the addition of an overall parity bit, it becomes the extended
Hamming code and can both detect and correct single-bit errors and detect (but not correct) double-bit errors.
1855:
1782:
551:
2031:
4099:
3871:
2285:
2260:
2991:
1716:
620:
to describe the system, including the number of data bits and error-correction bits in a block. For instance, parity includes a single bit for any data word, so assuming
568:
Another code in use at the time repeated every data bit multiple times in order to ensure that it was sent correctly. For instance, if the data bit to be sent is a 1, an
4637:
2882:
2853:
2705:
1670:
1445:
1406:
600:
errors but cannot correct them (the quantity of parity bits is even); at five bits, we can both detect and correct all two-bit errors, but not all three-bit errors.
729:
If a byte of data to be encoded is 10011010, then the data word (using _ to represent the parity bits) would be __1_001_1010, and the code word is 011100101010.
4331:
Finally, it can be shown that the minimum distance has increased from 3, in the code, to 4 in the code. Therefore, the code can be defined as
Hamming code.
3011:
686:
All bit positions that are powers of two (have a single 1 bit in the binary form of their position) are parity bits: 1, 2, 4, 8, etc. (1, 10, 100, 1000)
464:
A number of simple error-detecting codes were used before Hamming codes, but none were as effective as Hamming codes in the same overhead of space.
294:. Hamming codes can detect one-bit and two-bit errors, or correct one-bit errors without detection of uncorrected errors. By contrast, the simple
3847:
Note that H is not in standard form. To obtain G, elementary row operations can be used to obtain an equivalent matrix to H in systematic form:
4537:
2126:
616:
Hamming studied the existing coding schemes, including two-of-five, and generalized their concepts. To start with, he developed a
4713:
4305:
The addition of the fourth row effectively computes the sum of all the codeword bits (data and parity) as the fourth parity bit.
3385:
The same example from above with an extra parity bit. This diagram is not meant to correspond to the matrix H for this example.
265:
4626:
4661:
4567:
4486:
639:
Hamming also noticed the problems with flipping two or more bits, and described this as the "distance" (it is now called the
508:
A two-out-of-five code is an encoding scheme which uses five bits consisting of exactly three 0s and two 1s. This provides
692:
Each data bit is included in a unique set of 2 or more parity bits, as determined by the binary form of its bit position.
3389:
The Hamming code can easily be extended to an code by adding an extra parity bit on top of the (7,4) encoded word (see
723:
In general each parity bit covers all bits where the bitwise AND of the parity position and the bit position is non-zero.
2710:
732:
The choice of the parity, even or odd, is irrelevant but the same choice must be used for both encoding and decoding.
1876:
reliably detect triple bit errors) and resiliency (the ability to keep functioning in the face of single bit errors).
4599:
4435:
4375:
68:
46:
2010:{\displaystyle \mathbf {G} :={\begin{pmatrix}{\begin{array}{c|c}I_{k}&-A^{\text{T}}\\\end{array}}\end{pmatrix}}}
39:
1921:
Graphical depiction of the four data bits and three parity bits and which parity bits apply to which data bits
4512:
4365:
2887:
4718:
4703:
1788:
673:
conclude there were no corruptions, and otherwise, the index-XOR indicates the index of the corrupted bit.
405:, also known as a Simplex code. The parity-check matrix has the property that any two columns are pairwise
2091:{\displaystyle \mathbf {H} :={\begin{pmatrix}{\begin{array}{c|c}A&I_{n-k}\\\end{array}}\end{pmatrix}}}
4481:
Moon T. Error correction coding: Mathematical Methods and Algorithms. John Wiley and Sons, 2005.(Cap. 3)
4400:
4350:
2657:
Finally, these matrices can be mutated into equivalent non-systematic codes by the following operations:
1723:
4302:
The non-systematic form of G can be row reduced (using elementary row operations) to match this matrix.
511:
4723:
4591:
482:
that indicates whether the number of ones (bit-positions with values of one) in the preceding data was
258:
2052:
1967:
689:
All other bit positions, with two or more 1 bits in the binary form of their position, are data bits.
563:
322:
readers. In his original paper, Hamming elaborated his general idea, but specifically focused on the
4577:
4504:
33:
2268:
2243:
2937:
1676:
660:
to have the parity bits overlap, such that they managed to check each other as well as the data.
4430:, The Carus Mathematical Monographs (#21), Mathematical Association of America, pp. 16β17,
2858:
2829:
2681:
4708:
1636:
1411:
50:
4549:
494:
checking is poor, since it uses only a single bit, this method results in the least overhead.
4649:
1378:
298:
cannot correct errors, and can detect only an odd number of bits in error. Hamming codes are
251:
4581:
1465:
351:
152:
4688:
4611:
1997 International Symposium on Parallel Architectures, Algorithms and Networks (ISPAN '97)
4559:
4370:
503:
1460:
341:
307:
133:
8:
2263:
2170:
2099:
628:
code, with eight bits in total, of which seven are data. The repetition example would be
613:
potentially specify not only that an error occurred but also which bit caused the error.
406:
388:
318:
invented Hamming codes in 1950 as a way of automatically correcting errors introduced by
221:
4529:
4525:
2996:
283:
111:
311:
209:
197:
4657:
4595:
4563:
4482:
4431:
1365:
441:
126:
4533:
617:
4614:
4521:
4360:
2238:
2195:-tuples in the columns of matrix does not matter. The right hand side is just the (
1880:
641:
279:
4312:
is encoded (using the non-systematic form of G at the start of this section) into
1868:
single bit errors and therefore go undetected, unless no correction is attempted.
1473:
303:
168:
2227:
2204:
1879:
This extended Hamming code was popular in computer memory systems, starting with
1492:
576:
429:
315:
4683:
4678:
4618:
4697:
4355:
4345:
2664:
Elementary row operations (replacing a row with a linear combination of rows)
402:
636:
is the second number divided by the first, for our repetition example, 1/3.
4609:
D.K. Bhattacharryya, S. Nandi. "An efficient class of SEC-DED-AUED codes".
3390:
1926:
1512:
669:
479:
445:
323:
319:
299:
241:
4684:
CGI script for calculating Hamming distances (from R. Tervo, UNB, Canada)
483:
333:
terms, Hamming codes are a class of binary linear code. For each integer
330:
291:
4555:
4380:
487:
473:
453:
414:
295:
4587:
683:
Write the bit numbers in binary: 1, 10, 11, 100, 101, 110, 111, etc.
633:
444:
relay-based machine with cycle times in seconds. Input was fed in on
433:
398:
4427:
From Error-Correcting Codes through Sphere Packings to Simple Groups
1872:
error and at the same time detect (but not correct) a double error.
699:
significant bit set: bit 1 (the parity bit itself), 3, 5, 7, 9, etc.
2176:
of a Hamming code is constructed by listing all columns of length
2159:{\displaystyle \mathbf {H} \,\mathbf {G} ^{\text{T}}=\mathbf {0} }
391:
of a Hamming code is constructed by listing all columns of length
437:
1862:
3381:
1917:
1892:
90:
3365:
680:
Number the bits starting from 1: bit 1, 2, 3, 4, 5, 6, 7, etc.
621:
676:
An algorithm can be deduced from the following description:
4450:
4448:
4446:
3013:
from above, we have (after applying modulo 2, to the sum),
1895:
4627:"Mathematical Challenge April 2013 Error-correcting codes"
4608:
4405:
585:
code. This triple repetition code is a Hamming code with
4443:
2678:
From the above matrix we have 2 = 2 = 16 codewords. Let
720:
least significant bit set: bits 8β15, 24β31, 40β47, etc.
326:
code which adds three parity bits to four bits of data.
4460:
713:
least significant bit set: bits 4β7, 12β15, 20β23, etc.
3641:
3417:
3309:
3257:
3094:
3060:
2512:
2310:
2113:
in standard (or systematic) form. Regardless of form,
2048:
2017:
is called a (canonical) generator matrix of a linear (
1963:
4583:
Information Theory, Inference and Learning Algorithms
4084:
3856:
3626:
3402:
3021:
2999:
2940:
2890:
2861:
2832:
2713:
2684:
2497:
2295:
2271:
2246:
2129:
2034:
1949:
1791:
1726:
1679:
1639:
1414:
1381:
716:
Parity bit 8 covers all bit positions which have the
709:
Parity bit 4 covers all bit positions which have the
706:
least significant bit set: bits 2-3, 6-7, 10-11, etc.
702:
Parity bit 2 covers all bit positions which have the
695:
Parity bit 1 covers all bit positions which have the
514:
3393:). This can be summed up with the revised matrices:
624:
words with seven bits, Hamming described this as an
1408:can be covered. After discounting the parity bits,
4291:
4063:
3833:
3606:
3355:
3005:
2985:
2923:
2876:
2847:
2819:{\displaystyle {\vec {a}}=,\quad a_{i}\in \{0,1\}}
2818:
2699:
2647:
2479:
2279:
2254:
2187:is a matrix whose left side is all of the nonzero
2158:
2090:
2009:
1849:
1776:
1710:
1664:
1439:
1400:
545:
2218:by taking the transpose of the left hand side of
531:
518:
4695:
3370:Hamming code with an additional parity bit": -->
4648:Kythe, Dave K.; Kythe, Prem K. (28 July 2017).
1889:single error correction, double error detection
1451:varies, we get all the possible Hamming codes:
2931:where the summing operation is done modulo-2.
1863:Hamming codes with additional parity (SECDED)
302:, that is, they achieve the highest possible
259:
4505:"Error detecting and error correcting codes"
2813:
2801:
432:, the inventor of Hamming codes, worked at
4647:
4466:
4454:
3366:Hamming code with an additional parity bit
1938:
459:
266:
252:
2135:
735:This general rule can be shown visually:
69:Learn how and when to remove this message
4423:
3380:
2884:is given by the standard matrix product
2855:for any of the 16 possible data vectors
1916:
397:that are non-zero, which means that the
32:This article includes a list of general
4502:
4411:
497:
16:Family of linear error-correcting codes
4696:
4654:Algebraic and Stochastic Coding Theory
4576:
4477:
4475:
2924:{\displaystyle {\vec {x}}={\vec {a}}G}
2661:Column permutations (swapping columns)
2707:be a row vector of binary data bits,
1850:{\displaystyle (2^{m}-m-1)/(2^{m}-1)}
592:since there are two parity bits, and
364:. Hence the rate of Hamming codes is
4547:
2121:for linear block codes must satisfy
663:
18:
4679:Visual Explanation of Hamming Codes
4472:
1777:{\displaystyle (2^{m}-1,2^{m}-m-1)}
1350:
1343:
1336:
1329:
1322:
1270:
1263:
1256:
1249:
1242:
1235:
1228:
1221:
1195:
1180:
1173:
1166:
1159:
1144:
1137:
1130:
1123:
1103:
1096:
1085:
1078:
1067:
1060:
1049:
1042:
1031:
1024:
1008:
999:
990:
981:
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963:
954:
945:
936:
927:
13:
4526:10.1002/j.1538-7305.1950.tb00463.x
546:{\displaystyle {\binom {5}{3}}=10}
522:
38:it lacks sufficient corresponding
14:
4735:
4689:Tool for calculating Hamming code
4672:
4643:from the original on 2017-09-12.
4634:swissQuant Group Leadership Team
4543:from the original on 2022-10-09.
4503:Hamming, Richard Wesley (1950).
4086:
3858:
3628:
3404:
2499:
2297:
2273:
2248:
2180:that are pair-wise independent.
2152:
2138:
2131:
2036:
1951:
1901:
1447:bits remain for use as data. As
1351:
1344:
1337:
1330:
1323:
1271:
1264:
1257:
1250:
1243:
1236:
1229:
1222:
1196:
1181:
1174:
1167:
1160:
1145:
1138:
1131:
1124:
1104:
1097:
1086:
1079:
1068:
1061:
1050:
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1032:
1025:
1009:
1000:
991:
982:
973:
964:
955:
946:
937:
928:
632:, following the same logic. The
89:
23:
2787:
1375:parity bits, bits from 1 up to
4714:Error detection and correction
4656:. CRC Press. pp. 95β116.
4417:
4394:
3043:
3028:
2980:
2956:
2947:
2912:
2897:
2868:
2839:
2781:
2729:
2720:
2691:
2230:on the left hand side of
1887:(or SEC-DED, abbreviated from
1883:in 1961, where it is known as
1844:
1825:
1817:
1792:
1771:
1727:
607:
436:in the late 1940s on the Bell
1:
4513:Bell System Technical Journal
4495:
4376:ReedβSolomon error correction
4366:Low-density parity-check code
2993:. Using the generator matrix
557:
292:linear error-correcting codes
4424:Thompson, Thomas M. (1983),
2280:{\displaystyle \mathbf {H} }
2255:{\displaystyle \mathbf {G} }
2169:Since = = . The
2105:This is the construction of
95:The Hamming(7,4) code (with
7:
4338:
2986:{\displaystyle {\vec {a}}=}
2668:
2191:-tuples where order of the
1711:{\displaystyle k=2^{m}-m-1}
1352:
1345:
1338:
1331:
1324:
1272:
1265:
1258:
1251:
1244:
1237:
1230:
1223:
1197:
1182:
1175:
1168:
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1125:
1105:
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1080:
1069:
1062:
1051:
1044:
1033:
1026:
1010:
1001:
992:
983:
974:
965:
956:
947:
938:
929:
649:can detect but not correct
401:of the Hamming code is the
10:
4740:
4592:Cambridge University Press
2877:{\displaystyle {\vec {a}}}
2848:{\displaystyle {\vec {x}}}
2700:{\displaystyle {\vec {a}}}
1924:
561:
501:
471:
424:
340:there is a code-word with
4619:10.1109/ISPAN.1997.645128
1665:{\displaystyle n=2^{m}-1}
1623:
1440:{\displaystyle 2^{m}-m-1}
917:
852:
847:
742:
564:Triple modular redundancy
467:
247:
240:
235:
220:
208:
196:
167:
151:
132:
122:
117:
107:
88:
83:
4650:"Extended Hamming Codes"
4387:
4551:Error Correction Coding
2166:, an all-zeros matrix.
1939:Construction of G and H
1401:{\displaystyle 2^{m}-1}
460:Codes predating Hamming
403:shortened Hadamard code
53:more precise citations.
4548:Moon, Todd K. (2005).
4467:Kythe & Kythe 2017
4455:Kythe & Kythe 2017
4293:
4065:
3835:
3608:
3386:
3357:
3007:
2987:
2925:
2878:
2849:
2820:
2701:
2649:
2481:
2281:
2256:
2160:
2092:
2011:
1922:
1851:
1778:
1712:
1666:
1441:
1402:
547:
4560:John Wiley & Sons
4294:
4066:
3836:
3609:
3384:
3358:
3008:
2988:
2926:
2879:
2850:
2821:
2702:
2650:
2482:
2282:
2257:
2214:can be obtained from
2161:
2093:
2012:
1920:
1852:
1779:
1713:
1667:
1442:
1403:
548:
478:Parity adds a single
306:for codes with their
4613:. pp. 410β415.
4082:
3854:
3624:
3400:
3019:
2997:
2938:
2888:
2859:
2830:
2711:
2682:
2495:
2293:
2269:
2244:
2127:
2032:
1947:
1789:
1724:
1677:
1637:
1412:
1379:
512:
504:Two-out-of-five code
498:Two-out-of-five code
407:linearly independent
84:Binary Hamming codes
4719:Computer arithmetic
4704:American inventions
4414:, pp. 153β154.
2264:parity-check matrix
2171:parity-check matrix
2100:parity-check matrix
1364:errors, called the
389:parity-check matrix
4580:(September 2003).
4578:MacKay, David J.C.
4289:
4268:
4061:
4040:
3831:
3810:
3604:
3586:
3387:
3353:
3347:
3295:
3243:
3083:
3003:
2983:
2921:
2874:
2845:
2816:
2697:
2645:
2624:
2477:
2459:
2277:
2252:
2222:with the identity
2156:
2088:
2082:
2078:
2007:
2001:
1997:
1923:
1906:Hamming code": -->
1847:
1774:
1708:
1662:
1437:
1398:
853:Encoded data bits
543:
446:punched paper tape
316:Richard W. Hamming
284:telecommunications
112:Richard W. Hamming
4724:1951 in computing
4663:978-1-351-83245-8
4569:978-0-471-64800-0
4487:978-0-471-64800-0
3046:
3031:
3006:{\displaystyle G}
2950:
2934:For example, let
2915:
2900:
2871:
2842:
2723:
2694:
2145:
1992:
1860:
1859:
1359:
1358:
664:General algorithm
529:
442:electromechanical
276:
275:
127:Linear block code
79:
78:
71:
4731:
4667:
4644:
4642:
4631:
4622:
4605:
4573:
4544:
4542:
4509:
4489:
4479:
4470:
4464:
4458:
4452:
4441:
4440:
4421:
4415:
4409:
4403:
4398:
4371:ReedβMuller code
4361:Hamming distance
4327:
4324:
4321:
4318:
4315:
4311:
4298:
4296:
4295:
4290:
4285:
4284:
4273:
4269:
4089:
4070:
4068:
4067:
4062:
4057:
4056:
4045:
4041:
3861:
3840:
3838:
3837:
3832:
3827:
3826:
3815:
3814:
3631:
3613:
3611:
3610:
3605:
3603:
3602:
3591:
3590:
3407:
3378:
3377:
3373:
3362:
3360:
3359:
3354:
3352:
3351:
3300:
3299:
3248:
3247:
3088:
3087:
3048:
3047:
3039:
3033:
3032:
3024:
3012:
3010:
3009:
3004:
2992:
2990:
2989:
2984:
2952:
2951:
2943:
2930:
2928:
2927:
2922:
2917:
2916:
2908:
2902:
2901:
2893:
2883:
2881:
2880:
2875:
2873:
2872:
2864:
2854:
2852:
2851:
2846:
2844:
2843:
2835:
2825:
2823:
2822:
2817:
2797:
2796:
2780:
2779:
2767:
2766:
2754:
2753:
2741:
2740:
2725:
2724:
2716:
2706:
2704:
2703:
2698:
2696:
2695:
2687:
2654:
2652:
2651:
2646:
2641:
2640:
2629:
2628:
2502:
2486:
2484:
2483:
2478:
2476:
2475:
2464:
2463:
2300:
2286:
2284:
2283:
2278:
2276:
2261:
2259:
2258:
2253:
2251:
2239:generator matrix
2165:
2163:
2162:
2157:
2155:
2147:
2146:
2143:
2141:
2134:
2097:
2095:
2094:
2089:
2087:
2086:
2079:
2075:
2074:
2039:
2016:
2014:
2013:
2008:
2006:
2005:
1998:
1994:
1993:
1990:
1979:
1978:
1954:
1914:
1913:
1909:
1881:IBM 7030 Stretch
1856:
1854:
1853:
1848:
1837:
1836:
1824:
1804:
1803:
1783:
1781:
1780:
1775:
1758:
1757:
1739:
1738:
1717:
1715:
1714:
1709:
1695:
1694:
1671:
1669:
1668:
1663:
1655:
1654:
1631:
1619:502/511 β 0.982
1616:Hamming(511,502)
1602:247/255 β 0.969
1599:Hamming(255,247)
1585:120/127 β 0.945
1582:Hamming(127,120)
1454:
1453:
1450:
1446:
1444:
1443:
1438:
1424:
1423:
1407:
1405:
1404:
1399:
1391:
1390:
1374:
1355:
1354:
1348:
1347:
1341:
1340:
1334:
1333:
1327:
1326:
1275:
1274:
1268:
1267:
1261:
1260:
1254:
1253:
1247:
1246:
1240:
1239:
1233:
1232:
1226:
1225:
1200:
1199:
1185:
1184:
1178:
1177:
1171:
1170:
1164:
1163:
1149:
1148:
1142:
1141:
1135:
1134:
1128:
1127:
1108:
1107:
1101:
1100:
1090:
1089:
1083:
1082:
1072:
1071:
1065:
1064:
1054:
1053:
1047:
1046:
1036:
1035:
1029:
1028:
1013:
1012:
1004:
1003:
995:
994:
986:
985:
977:
976:
968:
967:
959:
958:
950:
949:
941:
940:
932:
931:
845:
840:
835:
830:
825:
820:
815:
810:
805:
800:
795:
790:
785:
780:
775:
770:
765:
760:
755:
750:
745:
740:
739:
655:
642:Hamming distance
595:
591:
583:error-correcting
574:
552:
550:
549:
544:
536:
535:
534:
521:
396:
386:
382:
363:
349:
339:
312:minimum distance
290:are a family of
280:computer science
268:
261:
254:
230:
216:
204:
192:
191:
189:
188:
185:
182:
163:
147:
140:
101:
93:
81:
80:
74:
67:
63:
60:
54:
49:this article by
40:inline citations
27:
26:
19:
4739:
4738:
4734:
4733:
4732:
4730:
4729:
4728:
4694:
4693:
4675:
4670:
4664:
4640:
4629:
4625:
4602:
4570:
4540:
4507:
4498:
4493:
4492:
4480:
4473:
4465:
4461:
4453:
4444:
4438:
4422:
4418:
4410:
4406:
4401:See Lemma 12 of
4399:
4395:
4390:
4385:
4341:
4325:
4322:
4319:
4316:
4313:
4309:
4274:
4267:
4266:
4261:
4256:
4251:
4246:
4241:
4236:
4231:
4225:
4224:
4219:
4214:
4209:
4204:
4199:
4194:
4189:
4183:
4182:
4177:
4172:
4167:
4162:
4157:
4152:
4147:
4141:
4140:
4135:
4130:
4125:
4120:
4115:
4110:
4105:
4098:
4094:
4093:
4085:
4083:
4080:
4079:
4046:
4039:
4038:
4033:
4028:
4023:
4018:
4013:
4008:
4003:
3997:
3996:
3991:
3986:
3981:
3976:
3971:
3966:
3961:
3955:
3954:
3949:
3944:
3939:
3934:
3929:
3924:
3919:
3913:
3912:
3907:
3902:
3897:
3892:
3887:
3882:
3877:
3870:
3866:
3865:
3857:
3855:
3852:
3851:
3844:
3816:
3809:
3808:
3803:
3798:
3793:
3788:
3783:
3778:
3773:
3767:
3766:
3761:
3756:
3751:
3746:
3741:
3736:
3731:
3725:
3724:
3719:
3714:
3709:
3704:
3699:
3694:
3689:
3683:
3682:
3677:
3672:
3667:
3662:
3657:
3652:
3647:
3637:
3636:
3635:
3627:
3625:
3622:
3621:
3592:
3585:
3584:
3579:
3574:
3569:
3564:
3559:
3554:
3549:
3543:
3542:
3537:
3532:
3527:
3522:
3517:
3512:
3507:
3501:
3500:
3495:
3490:
3485:
3480:
3475:
3470:
3465:
3459:
3458:
3453:
3448:
3443:
3438:
3433:
3428:
3423:
3413:
3412:
3411:
3403:
3401:
3398:
3397:
3379:
3375:
3371:
3369:
3368:
3346:
3345:
3340:
3335:
3330:
3325:
3320:
3315:
3305:
3304:
3294:
3293:
3288:
3283:
3278:
3273:
3268:
3263:
3253:
3252:
3242:
3241:
3236:
3231:
3226:
3221:
3216:
3211:
3205:
3204:
3199:
3194:
3189:
3184:
3179:
3174:
3168:
3167:
3162:
3157:
3152:
3147:
3142:
3137:
3131:
3130:
3125:
3120:
3115:
3110:
3105:
3100:
3090:
3089:
3082:
3081:
3076:
3071:
3066:
3056:
3055:
3038:
3037:
3023:
3022:
3020:
3017:
3016:
2998:
2995:
2994:
2942:
2941:
2939:
2936:
2935:
2907:
2906:
2892:
2891:
2889:
2886:
2885:
2863:
2862:
2860:
2857:
2856:
2834:
2833:
2831:
2828:
2827:
2826:. The codeword
2792:
2788:
2775:
2771:
2762:
2758:
2749:
2745:
2736:
2732:
2715:
2714:
2712:
2709:
2708:
2686:
2685:
2683:
2680:
2679:
2671:
2630:
2623:
2622:
2617:
2612:
2607:
2602:
2597:
2592:
2586:
2585:
2580:
2575:
2570:
2565:
2560:
2555:
2549:
2548:
2543:
2538:
2533:
2528:
2523:
2518:
2508:
2507:
2506:
2498:
2496:
2493:
2492:
2465:
2458:
2457:
2452:
2447:
2442:
2437:
2432:
2427:
2421:
2420:
2415:
2410:
2405:
2400:
2395:
2390:
2384:
2383:
2378:
2373:
2368:
2363:
2358:
2353:
2347:
2346:
2341:
2336:
2331:
2326:
2321:
2316:
2306:
2305:
2304:
2296:
2294:
2291:
2290:
2272:
2270:
2267:
2266:
2247:
2245:
2242:
2241:
2228:identity matrix
2205:identity matrix
2151:
2142:
2137:
2136:
2130:
2128:
2125:
2124:
2081:
2080:
2077:
2076:
2064:
2060:
2058:
2051:
2044:
2043:
2035:
2033:
2030:
2029:
2000:
1999:
1996:
1995:
1989:
1985:
1980:
1974:
1970:
1966:
1959:
1958:
1950:
1948:
1945:
1944:
1941:
1929:
1915:
1911:
1907:
1905:
1904:
1865:
1832:
1828:
1820:
1799:
1795:
1790:
1787:
1786:
1753:
1749:
1734:
1730:
1725:
1722:
1721:
1690:
1686:
1678:
1675:
1674:
1650:
1646:
1638:
1635:
1634:
1629:
1493:repetition code
1490:
1448:
1419:
1415:
1413:
1410:
1409:
1386:
1382:
1380:
1377:
1376:
1372:
921:
919:
843:
838:
833:
828:
823:
818:
813:
808:
803:
798:
793:
788:
783:
778:
773:
768:
763:
758:
753:
748:
743:
666:
650:
610:
593:
586:
577:repetition code
569:
566:
560:
530:
517:
516:
515:
513:
510:
509:
506:
500:
476:
470:
462:
430:Richard Hamming
427:
392:
384:
365:
354:
344:
334:
272:
229:
226:
214:
202:
186:
183:
178:
177:
175:
173:
157:
142:
138:
103:
96:
75:
64:
58:
55:
45:Please help to
44:
28:
24:
17:
12:
11:
5:
4737:
4727:
4726:
4721:
4716:
4711:
4706:
4692:
4691:
4686:
4681:
4674:
4673:External links
4671:
4669:
4668:
4662:
4645:
4636:. April 2013.
4623:
4606:
4600:
4574:
4568:
4545:
4520:(2): 147β160.
4499:
4497:
4494:
4491:
4490:
4471:
4459:
4457:, p. 115.
4442:
4436:
4416:
4412:Hamming (1950)
4404:
4392:
4391:
4389:
4386:
4384:
4383:
4378:
4373:
4368:
4363:
4358:
4353:
4348:
4342:
4340:
4337:
4300:
4299:
4288:
4283:
4280:
4277:
4272:
4265:
4262:
4260:
4257:
4255:
4252:
4250:
4247:
4245:
4242:
4240:
4237:
4235:
4232:
4230:
4227:
4226:
4223:
4220:
4218:
4215:
4213:
4210:
4208:
4205:
4203:
4200:
4198:
4195:
4193:
4190:
4188:
4185:
4184:
4181:
4178:
4176:
4173:
4171:
4168:
4166:
4163:
4161:
4158:
4156:
4153:
4151:
4148:
4146:
4143:
4142:
4139:
4136:
4134:
4131:
4129:
4126:
4124:
4121:
4119:
4116:
4114:
4111:
4109:
4106:
4104:
4101:
4100:
4097:
4092:
4088:
4072:
4071:
4060:
4055:
4052:
4049:
4044:
4037:
4034:
4032:
4029:
4027:
4024:
4022:
4019:
4017:
4014:
4012:
4009:
4007:
4004:
4002:
3999:
3998:
3995:
3992:
3990:
3987:
3985:
3982:
3980:
3977:
3975:
3972:
3970:
3967:
3965:
3962:
3960:
3957:
3956:
3953:
3950:
3948:
3945:
3943:
3940:
3938:
3935:
3933:
3930:
3928:
3925:
3923:
3920:
3918:
3915:
3914:
3911:
3908:
3906:
3903:
3901:
3898:
3896:
3893:
3891:
3888:
3886:
3883:
3881:
3878:
3876:
3873:
3872:
3869:
3864:
3860:
3842:
3841:
3830:
3825:
3822:
3819:
3813:
3807:
3804:
3802:
3799:
3797:
3794:
3792:
3789:
3787:
3784:
3782:
3779:
3777:
3774:
3772:
3769:
3768:
3765:
3762:
3760:
3757:
3755:
3752:
3750:
3747:
3745:
3742:
3740:
3737:
3735:
3732:
3730:
3727:
3726:
3723:
3720:
3718:
3715:
3713:
3710:
3708:
3705:
3703:
3700:
3698:
3695:
3693:
3690:
3688:
3685:
3684:
3681:
3678:
3676:
3673:
3671:
3668:
3666:
3663:
3661:
3658:
3656:
3653:
3651:
3648:
3646:
3643:
3642:
3640:
3634:
3630:
3615:
3614:
3601:
3598:
3595:
3589:
3583:
3580:
3578:
3575:
3573:
3570:
3568:
3565:
3563:
3560:
3558:
3555:
3553:
3550:
3548:
3545:
3544:
3541:
3538:
3536:
3533:
3531:
3528:
3526:
3523:
3521:
3518:
3516:
3513:
3511:
3508:
3506:
3503:
3502:
3499:
3496:
3494:
3491:
3489:
3486:
3484:
3481:
3479:
3476:
3474:
3471:
3469:
3466:
3464:
3461:
3460:
3457:
3454:
3452:
3449:
3447:
3444:
3442:
3439:
3437:
3434:
3432:
3429:
3427:
3424:
3422:
3419:
3418:
3416:
3410:
3406:
3367:
3364:
3350:
3344:
3341:
3339:
3336:
3334:
3331:
3329:
3326:
3324:
3321:
3319:
3316:
3314:
3311:
3310:
3308:
3303:
3298:
3292:
3289:
3287:
3284:
3282:
3279:
3277:
3274:
3272:
3269:
3267:
3264:
3262:
3259:
3258:
3256:
3251:
3246:
3240:
3237:
3235:
3232:
3230:
3227:
3225:
3222:
3220:
3217:
3215:
3212:
3210:
3207:
3206:
3203:
3200:
3198:
3195:
3193:
3190:
3188:
3185:
3183:
3180:
3178:
3175:
3173:
3170:
3169:
3166:
3163:
3161:
3158:
3156:
3153:
3151:
3148:
3146:
3143:
3141:
3138:
3136:
3133:
3132:
3129:
3126:
3124:
3121:
3119:
3116:
3114:
3111:
3109:
3106:
3104:
3101:
3099:
3096:
3095:
3093:
3086:
3080:
3077:
3075:
3072:
3070:
3067:
3065:
3062:
3061:
3059:
3054:
3051:
3045:
3042:
3036:
3030:
3027:
3002:
2982:
2979:
2976:
2973:
2970:
2967:
2964:
2961:
2958:
2955:
2949:
2946:
2920:
2914:
2911:
2905:
2899:
2896:
2870:
2867:
2841:
2838:
2815:
2812:
2809:
2806:
2803:
2800:
2795:
2791:
2786:
2783:
2778:
2774:
2770:
2765:
2761:
2757:
2752:
2748:
2744:
2739:
2735:
2731:
2728:
2722:
2719:
2693:
2690:
2676:
2675:
2670:
2667:
2666:
2665:
2662:
2644:
2639:
2636:
2633:
2627:
2621:
2618:
2616:
2613:
2611:
2608:
2606:
2603:
2601:
2598:
2596:
2593:
2591:
2588:
2587:
2584:
2581:
2579:
2576:
2574:
2571:
2569:
2566:
2564:
2561:
2559:
2556:
2554:
2551:
2550:
2547:
2544:
2542:
2539:
2537:
2534:
2532:
2529:
2527:
2524:
2522:
2519:
2517:
2514:
2513:
2511:
2505:
2501:
2474:
2471:
2468:
2462:
2456:
2453:
2451:
2448:
2446:
2443:
2441:
2438:
2436:
2433:
2431:
2428:
2426:
2423:
2422:
2419:
2416:
2414:
2411:
2409:
2406:
2404:
2401:
2399:
2396:
2394:
2391:
2389:
2386:
2385:
2382:
2379:
2377:
2374:
2372:
2369:
2367:
2364:
2362:
2359:
2357:
2354:
2352:
2349:
2348:
2345:
2342:
2340:
2337:
2335:
2332:
2330:
2327:
2325:
2322:
2320:
2317:
2315:
2312:
2311:
2309:
2303:
2299:
2275:
2250:
2154:
2150:
2140:
2133:
2085:
2073:
2070:
2067:
2063:
2059:
2057:
2054:
2053:
2050:
2049:
2047:
2042:
2038:
2004:
1988:
1984:
1981:
1977:
1973:
1969:
1968:
1965:
1964:
1962:
1957:
1953:
1940:
1937:
1925:Main article:
1903:
1900:
1864:
1861:
1858:
1857:
1846:
1843:
1840:
1835:
1831:
1827:
1823:
1819:
1816:
1813:
1810:
1807:
1802:
1798:
1794:
1784:
1773:
1770:
1767:
1764:
1761:
1756:
1752:
1748:
1745:
1742:
1737:
1733:
1729:
1718:
1707:
1704:
1701:
1698:
1693:
1689:
1685:
1682:
1672:
1661:
1658:
1653:
1649:
1645:
1642:
1632:
1626:
1625:
1621:
1620:
1617:
1614:
1611:
1608:
1604:
1603:
1600:
1597:
1594:
1591:
1587:
1586:
1583:
1580:
1577:
1574:
1570:
1569:
1568:57/63 β 0.905
1566:
1565:Hamming(63,57)
1563:
1560:
1557:
1553:
1552:
1551:26/31 β 0.839
1549:
1548:Hamming(31,26)
1546:
1543:
1540:
1536:
1535:
1534:11/15 β 0.733
1532:
1531:Hamming(15,11)
1529:
1526:
1523:
1519:
1518:
1515:
1510:
1507:
1504:
1500:
1499:
1496:
1487:
1484:
1481:
1477:
1476:
1471:
1468:
1463:
1458:
1436:
1433:
1430:
1427:
1422:
1418:
1397:
1394:
1389:
1385:
1366:error syndrome
1361:
1360:
1357:
1356:
1349:
1342:
1335:
1328:
1321:
1319:
1317:
1315:
1313:
1311:
1309:
1307:
1305:
1303:
1301:
1299:
1297:
1295:
1293:
1291:
1287:
1286:
1284:
1282:
1280:
1278:
1276:
1269:
1262:
1255:
1248:
1241:
1234:
1227:
1220:
1218:
1216:
1214:
1212:
1210:
1208:
1206:
1202:
1201:
1194:
1192:
1190:
1188:
1186:
1179:
1172:
1165:
1158:
1156:
1154:
1152:
1150:
1143:
1136:
1129:
1122:
1120:
1118:
1116:
1112:
1111:
1109:
1102:
1095:
1093:
1091:
1084:
1077:
1075:
1073:
1066:
1059:
1057:
1055:
1048:
1041:
1039:
1037:
1030:
1023:
1021:
1017:
1016:
1014:
1007:
1005:
998:
996:
989:
987:
980:
978:
971:
969:
962:
960:
953:
951:
944:
942:
935:
933:
926:
923:
915:
914:
911:
908:
905:
902:
899:
896:
893:
890:
887:
884:
881:
878:
875:
872:
869:
866:
863:
860:
857:
854:
850:
849:
846:
841:
836:
831:
826:
821:
816:
811:
806:
801:
796:
791:
786:
781:
776:
771:
766:
761:
756:
751:
746:
727:
726:
725:
724:
721:
714:
707:
700:
690:
687:
684:
681:
665:
662:
609:
606:
562:Main article:
559:
556:
542:
539:
533:
528:
525:
520:
502:Main article:
499:
496:
472:Main article:
469:
466:
461:
458:
426:
423:
352:message length
274:
273:
271:
270:
263:
256:
248:
245:
244:
238:
237:
233:
232:
227:
224:
218:
217:
212:
206:
205:
200:
194:
193:
171:
165:
164:
155:
153:Message length
149:
148:
136:
130:
129:
124:
120:
119:
118:Classification
115:
114:
109:
105:
104:
94:
86:
85:
77:
76:
31:
29:
22:
15:
9:
6:
4:
3:
2:
4736:
4725:
4722:
4720:
4717:
4715:
4712:
4710:
4709:Coding theory
4707:
4705:
4702:
4701:
4699:
4690:
4687:
4685:
4682:
4680:
4677:
4676:
4665:
4659:
4655:
4651:
4646:
4639:
4635:
4628:
4624:
4620:
4616:
4612:
4607:
4603:
4601:0-521-64298-1
4597:
4593:
4589:
4585:
4584:
4579:
4575:
4571:
4565:
4561:
4557:
4553:
4552:
4546:
4539:
4535:
4531:
4527:
4523:
4519:
4515:
4514:
4506:
4501:
4500:
4488:
4484:
4478:
4476:
4469:, p. 95.
4468:
4463:
4456:
4451:
4449:
4447:
4439:
4437:0-88385-023-0
4433:
4429:
4428:
4420:
4413:
4408:
4402:
4397:
4393:
4382:
4379:
4377:
4374:
4372:
4369:
4367:
4364:
4362:
4359:
4357:
4356:Hamming bound
4354:
4352:
4349:
4347:
4346:Coding theory
4344:
4343:
4336:
4332:
4329:
4308:For example,
4306:
4303:
4286:
4281:
4278:
4275:
4270:
4263:
4258:
4253:
4248:
4243:
4238:
4233:
4228:
4221:
4216:
4211:
4206:
4201:
4196:
4191:
4186:
4179:
4174:
4169:
4164:
4159:
4154:
4149:
4144:
4137:
4132:
4127:
4122:
4117:
4112:
4107:
4102:
4095:
4090:
4078:
4077:
4076:
4058:
4053:
4050:
4047:
4042:
4035:
4030:
4025:
4020:
4015:
4010:
4005:
4000:
3993:
3988:
3983:
3978:
3973:
3968:
3963:
3958:
3951:
3946:
3941:
3936:
3931:
3926:
3921:
3916:
3909:
3904:
3899:
3894:
3889:
3884:
3879:
3874:
3867:
3862:
3850:
3849:
3848:
3845:
3828:
3823:
3820:
3817:
3811:
3805:
3800:
3795:
3790:
3785:
3780:
3775:
3770:
3763:
3758:
3753:
3748:
3743:
3738:
3733:
3728:
3721:
3716:
3711:
3706:
3701:
3696:
3691:
3686:
3679:
3674:
3669:
3664:
3659:
3654:
3649:
3644:
3638:
3632:
3620:
3619:
3618:
3599:
3596:
3593:
3587:
3581:
3576:
3571:
3566:
3561:
3556:
3551:
3546:
3539:
3534:
3529:
3524:
3519:
3514:
3509:
3504:
3497:
3492:
3487:
3482:
3477:
3472:
3467:
3462:
3455:
3450:
3445:
3440:
3435:
3430:
3425:
3420:
3414:
3408:
3396:
3395:
3394:
3392:
3383:
3374:
3363:
3348:
3342:
3337:
3332:
3327:
3322:
3317:
3312:
3306:
3301:
3296:
3290:
3285:
3280:
3275:
3270:
3265:
3260:
3254:
3249:
3244:
3238:
3233:
3228:
3223:
3218:
3213:
3208:
3201:
3196:
3191:
3186:
3181:
3176:
3171:
3164:
3159:
3154:
3149:
3144:
3139:
3134:
3127:
3122:
3117:
3112:
3107:
3102:
3097:
3091:
3084:
3078:
3073:
3068:
3063:
3057:
3052:
3049:
3040:
3034:
3025:
3014:
3000:
2977:
2974:
2971:
2968:
2965:
2962:
2959:
2953:
2944:
2932:
2918:
2909:
2903:
2894:
2865:
2836:
2810:
2807:
2804:
2798:
2793:
2789:
2784:
2776:
2772:
2768:
2763:
2759:
2755:
2750:
2746:
2742:
2737:
2733:
2726:
2717:
2688:
2673:
2672:
2663:
2660:
2659:
2658:
2655:
2642:
2637:
2634:
2631:
2625:
2619:
2614:
2609:
2604:
2599:
2594:
2589:
2582:
2577:
2572:
2567:
2562:
2557:
2552:
2545:
2540:
2535:
2530:
2525:
2520:
2515:
2509:
2503:
2490:
2487:
2472:
2469:
2466:
2460:
2454:
2449:
2444:
2439:
2434:
2429:
2424:
2417:
2412:
2407:
2402:
2397:
2392:
2387:
2380:
2375:
2370:
2365:
2360:
2355:
2350:
2343:
2338:
2333:
2328:
2323:
2318:
2313:
2307:
2301:
2288:
2265:
2240:
2235:
2233:
2229:
2225:
2221:
2217:
2213:
2208:
2206:
2202:
2199: β
2198:
2194:
2190:
2186:
2181:
2179:
2175:
2172:
2167:
2148:
2122:
2120:
2116:
2112:
2108:
2103:
2101:
2083:
2071:
2068:
2065:
2061:
2055:
2045:
2040:
2026:
2024:
2020:
2002:
1986:
1982:
1975:
1971:
1960:
1955:
1936:
1933:
1928:
1919:
1910:
1899:
1897:
1894:
1890:
1886:
1882:
1877:
1873:
1869:
1841:
1838:
1833:
1829:
1821:
1814:
1811:
1808:
1805:
1800:
1796:
1785:
1768:
1765:
1762:
1759:
1754:
1750:
1746:
1743:
1740:
1735:
1731:
1719:
1705:
1702:
1699:
1696:
1691:
1687:
1683:
1680:
1673:
1659:
1656:
1651:
1647:
1643:
1640:
1633:
1628:
1627:
1622:
1618:
1615:
1612:
1609:
1606:
1605:
1601:
1598:
1595:
1592:
1589:
1588:
1584:
1581:
1578:
1575:
1572:
1571:
1567:
1564:
1561:
1558:
1555:
1554:
1550:
1547:
1544:
1541:
1538:
1537:
1533:
1530:
1527:
1524:
1521:
1520:
1516:
1514:
1511:
1508:
1505:
1502:
1501:
1497:
1494:
1488:
1485:
1482:
1479:
1478:
1475:
1472:
1469:
1467:
1464:
1462:
1459:
1456:
1455:
1452:
1434:
1431:
1428:
1425:
1420:
1416:
1395:
1392:
1387:
1383:
1369:
1367:
1320:
1318:
1316:
1314:
1312:
1310:
1308:
1306:
1304:
1302:
1300:
1298:
1296:
1294:
1292:
1289:
1288:
1285:
1283:
1281:
1279:
1277:
1219:
1217:
1215:
1213:
1211:
1209:
1207:
1204:
1203:
1193:
1191:
1189:
1187:
1157:
1155:
1153:
1151:
1121:
1119:
1117:
1114:
1113:
1110:
1094:
1092:
1076:
1074:
1058:
1056:
1040:
1038:
1022:
1019:
1018:
1015:
1006:
997:
988:
979:
970:
961:
952:
943:
934:
924:
916:
912:
909:
906:
903:
900:
897:
894:
891:
888:
885:
882:
879:
876:
873:
870:
867:
864:
861:
858:
855:
851:
842:
837:
832:
827:
822:
817:
812:
807:
802:
797:
792:
787:
782:
777:
772:
767:
762:
757:
752:
747:
741:
738:
737:
736:
733:
730:
722:
719:
715:
712:
708:
705:
701:
698:
694:
693:
691:
688:
685:
682:
679:
678:
677:
674:
671:
661:
657:
653:
648:
644:
643:
637:
635:
631:
627:
623:
619:
614:
605:
601:
597:
594:2 β 2 β 1 = 1
589:
584:
579:
578:
572:
565:
555:
540:
537:
526:
523:
505:
495:
491:
489:
485:
481:
475:
465:
457:
455:
449:
447:
443:
440:computer, an
439:
435:
431:
422:
420:
416:
410:
408:
404:
400:
395:
390:
380:
376:
372:
368:
361:
357:
353:
347:
343:
337:
332:
327:
325:
321:
317:
313:
309:
305:
301:
300:perfect codes
297:
293:
289:
288:Hamming codes
285:
281:
269:
264:
262:
257:
255:
250:
249:
246:
243:
239:
234:
225:
223:
219:
213:
211:
210:Alphabet size
207:
201:
199:
195:
181:
172:
170:
166:
161:
156:
154:
150:
145:
137:
135:
131:
128:
125:
121:
116:
113:
110:
106:
99:
92:
87:
82:
73:
70:
62:
52:
48:
42:
41:
35:
30:
21:
20:
4653:
4610:
4582:
4550:
4517:
4511:
4462:
4426:
4419:
4407:
4396:
4333:
4330:
4307:
4304:
4301:
4073:
3846:
3843:
3616:
3391:Hamming(7,4)
3388:
3015:
2933:
2677:
2656:
2491:
2488:
2289:
2236:
2231:
2223:
2219:
2215:
2211:
2209:
2200:
2196:
2192:
2188:
2184:
2182:
2177:
2173:
2168:
2123:
2118:
2114:
2110:
2106:
2104:
2098:is called a
2027:
2022:
2018:
1943:The matrix
1942:
1934:
1930:
1927:Hamming(7,4)
1902:Hamming code
1888:
1884:
1878:
1874:
1870:
1866:
1517:4/7 β 0.571
1513:Hamming(7,4)
1498:1/3 β 0.333
1489:Hamming(3,1)
1370:
1362:
744:Bit position
734:
731:
728:
717:
710:
703:
696:
675:
667:
658:
651:
646:
640:
638:
629:
625:
618:nomenclature
615:
611:
602:
598:
587:
582:
575:
570:
567:
507:
492:
477:
463:
450:
428:
418:
411:
393:
378:
374:
370:
366:
359:
355:
345:
342:block length
335:
331:mathematical
328:
324:Hamming(7,4)
320:punched card
308:block length
287:
277:
242:perfect code
179:
159:
143:
134:Block length
97:
65:
56:
37:
1457:Parity bits
608:Description
296:parity code
108:Named after
51:introducing
4698:Categories
4556:New Jersey
4496:References
4381:Turbo code
4351:Golay code
1898:families.
1461:Total bits
596:data bit.
558:Repetition
474:Parity bit
454:ECC memory
415:ECC memory
314:of three.
236:Properties
59:March 2013
34:references
4588:Cambridge
3044:→
3029:→
2948:→
2913:→
2898:→
2869:→
2840:→
2799:∈
2721:→
2692:→
2237:The code
2069:−
1983:−
1839:−
1812:−
1806:−
1766:−
1760:−
1741:−
1703:−
1697:−
1657:−
1466:Data bits
1432:−
1426:−
1393:−
922:coverage
634:code rate
434:Bell Labs
413:ECC RAM (
399:dual code
381:/ (2 β 1)
4638:Archived
4538:Archived
4534:61141773
4339:See also
2669:Encoding
2262:and the
2025:) code,
1491:(Triple
656:errors.
222:Notation
198:Distance
2674:Example
1720:Hamming
438:Model V
425:History
348:= 2 β 1
190:
187:(2 β 1)
176:
47:improve
4660:
4598:
4566:
4532:
4485:
4434:
1893:Xilinx
1885:SECDED
918:Parity
718:fourth
704:second
468:Parity
419:SECDED
387:. The
377:= 1 β
358:= 2 β
141:where
36:, but
4641:(PDF)
4630:(PDF)
4541:(PDF)
4530:S2CID
4508:(PDF)
4388:Notes
2287:are:
2183:Thus
1371:With
711:third
697:least
630:(3,1)
626:(8,7)
622:ASCII
385:2 β 1
231:-code
139:2 β 1
4658:ISBN
4596:ISBN
4564:ISBN
4483:ISBN
4432:ISBN
4310:1011
3617:and
3372:edit
2489:and
2117:and
2109:and
2028:and
1908:edit
1896:FPGA
1624:...
1474:Rate
1470:Name
1290:p16
913:d15
898:d11
848:...
590:= 2,
484:even
350:and
310:and
304:rate
282:and
174:1 β
169:Rate
158:2 β
123:Type
4615:doi
4522:doi
4323:011
2210:So
1613:502
1610:511
1596:247
1593:255
1579:120
1576:127
1205:p8
1115:p4
1020:p2
925:p1
920:bit
910:d14
907:d13
904:d12
901:p16
895:d10
874:d4
862:d1
856:p1
670:XOR
654:β 1
573:= 3
488:odd
486:or
480:bit
362:β 1
338:β₯ 2
329:In
278:In
162:β 1
146:β₯ 2
100:= 3
4700::
4652:.
4632:.
4594:.
4590::
4586:.
4562:.
4558::
4554:.
4536:.
4528:.
4518:29
4516:.
4510:.
4474:^
4445:^
4314:01
3633::=
3409::=
2504::=
2302::=
2234:.
2207:.
2203:)-
2102:.
2041::=
1956::=
1562:57
1559:63
1545:26
1542:31
1528:11
1525:15
892:d9
889:d8
886:d7
883:d6
880:d5
877:p8
871:d3
868:d2
865:p4
859:p2
844:20
839:19
834:18
829:17
824:16
819:15
814:14
809:13
804:12
799:11
794:10
541:10
456:.
421:.
409:.
373:/
369:=
286:,
4666:.
4621:.
4617::
4604:.
4572:.
4524::
4326:0
4320:0
4317:1
4287:.
4282:8
4279:,
4276:4
4271:)
4264:0
4259:1
4254:1
4249:1
4244:1
4239:0
4234:0
4229:0
4222:1
4217:0
4212:1
4207:1
4202:0
4197:1
4192:0
4187:0
4180:1
4175:1
4170:0
4165:1
4160:0
4155:0
4150:1
4145:0
4138:1
4133:1
4128:1
4123:0
4118:0
4113:0
4108:0
4103:1
4096:(
4091:=
4087:G
4059:.
4054:8
4051:,
4048:4
4043:)
4036:1
4031:0
4026:0
4021:0
4016:0
4011:1
4006:1
4001:1
3994:0
3989:1
3984:0
3979:0
3974:1
3969:0
3964:1
3959:1
3952:0
3947:0
3942:1
3937:0
3932:1
3927:1
3922:0
3917:1
3910:0
3905:0
3900:0
3895:1
3890:1
3885:1
3880:1
3875:0
3868:(
3863:=
3859:H
3829:.
3824:8
3821:,
3818:4
3812:)
3806:1
3801:1
3796:1
3791:1
3786:1
3781:1
3776:1
3771:1
3764:0
3759:1
3754:1
3749:1
3744:1
3739:0
3734:0
3729:0
3722:0
3717:1
3712:1
3707:0
3702:0
3697:1
3692:1
3687:0
3680:0
3675:1
3670:0
3665:1
3660:0
3655:1
3650:0
3645:1
3639:(
3629:H
3600:8
3597:,
3594:4
3588:)
3582:0
3577:1
3572:0
3567:0
3562:1
3557:0
3552:1
3547:1
3540:1
3535:0
3530:1
3525:0
3520:1
3515:0
3510:1
3505:0
3498:1
3493:0
3488:0
3483:1
3478:1
3473:0
3468:0
3463:1
3456:1
3451:0
3446:0
3441:0
3436:0
3431:1
3426:1
3421:1
3415:(
3405:G
3376:]
3349:)
3343:0
3338:1
3333:0
3328:1
3323:1
3318:0
3313:1
3307:(
3302:=
3297:)
3291:2
3286:3
3281:2
3276:1
3271:1
3266:0
3261:1
3255:(
3250:=
3245:)
3239:1
3234:1
3229:1
3224:1
3219:0
3214:0
3209:0
3202:1
3197:1
3192:0
3187:0
3182:1
3177:0
3172:0
3165:1
3160:0
3155:1
3150:0
3145:0
3140:1
3135:0
3128:0
3123:1
3118:1
3113:0
3108:0
3103:0
3098:1
3092:(
3085:)
3079:1
3074:1
3069:0
3064:1
3058:(
3053:=
3050:G
3041:a
3035:=
3026:x
3001:G
2981:]
2978:1
2975:,
2972:1
2969:,
2966:0
2963:,
2960:1
2957:[
2954:=
2945:a
2919:G
2910:a
2904:=
2895:x
2866:a
2837:x
2814:}
2811:1
2808:,
2805:0
2802:{
2794:i
2790:a
2785:,
2782:]
2777:4
2773:a
2769:,
2764:3
2760:a
2756:,
2751:2
2747:a
2743:,
2738:1
2734:a
2730:[
2727:=
2718:a
2689:a
2643:.
2638:7
2635:,
2632:3
2626:)
2620:1
2615:0
2610:0
2605:1
2600:1
2595:1
2590:0
2583:0
2578:1
2573:0
2568:1
2563:1
2558:0
2553:1
2546:0
2541:0
2536:1
2531:1
2526:0
2521:1
2516:1
2510:(
2500:H
2473:7
2470:,
2467:4
2461:)
2455:1
2450:1
2445:1
2440:1
2435:0
2430:0
2425:0
2418:1
2413:1
2408:0
2403:0
2398:1
2393:0
2388:0
2381:1
2376:0
2371:1
2366:0
2361:0
2356:1
2351:0
2344:0
2339:1
2334:1
2329:0
2324:0
2319:0
2314:1
2308:(
2298:G
2274:H
2249:G
2232:G
2226:-
2224:k
2220:H
2216:H
2212:G
2201:k
2197:n
2193:n
2189:n
2185:H
2178:m
2174:H
2153:0
2149:=
2144:T
2139:G
2132:H
2119:H
2115:G
2111:H
2107:G
2084:)
2072:k
2066:n
2062:I
2056:A
2046:(
2037:H
2023:k
2021:,
2019:n
2003:)
1991:T
1987:A
1976:k
1972:I
1961:(
1952:G
1912:]
1845:)
1842:1
1834:m
1830:2
1826:(
1822:/
1818:)
1815:1
1809:m
1801:m
1797:2
1793:(
1772:)
1769:1
1763:m
1755:m
1751:2
1747:,
1744:1
1736:m
1732:2
1728:(
1706:1
1700:m
1692:m
1688:2
1684:=
1681:k
1660:1
1652:m
1648:2
1644:=
1641:n
1630:m
1607:9
1590:8
1573:7
1556:6
1539:5
1522:4
1509:4
1506:7
1503:3
1495:)
1486:1
1483:3
1480:2
1449:m
1435:1
1429:m
1421:m
1417:2
1396:1
1388:m
1384:2
1373:m
789:9
784:8
779:7
774:6
769:5
764:4
759:3
754:2
749:1
652:k
647:k
588:m
571:n
538:=
532:)
527:3
524:5
519:(
394:r
379:r
375:n
371:k
367:R
360:r
356:k
346:n
336:r
267:e
260:t
253:v
228:2
215:2
203:3
184:/
180:r
160:r
144:r
102:)
98:r
72:)
66:(
61:)
57:(
43:.
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