878:
650:
873:{\displaystyle {\begin{aligned}\mathbb {P} (x{\mbox{ received}}\mid y{\mbox{ sent}})&{}={\frac {\mathbb {P} (x{\mbox{ received}},y{\mbox{ sent}})}{\mathbb {P} (y{\mbox{ sent}})}}\\&{}=\mathbb {P} (y{\mbox{ sent}}\mid x{\mbox{ received}})\cdot {\frac {\mathbb {P} (x{\mbox{ received}})}{\mathbb {P} (y{\mbox{ sent}})}}.\end{aligned}}}
1587:
403:
Each codeword does not have an expected possibility: there may be more than one codeword with an equal likelihood of mutating into the received message. In such a case, the sender and receiver(s) must agree ahead of time on a decoding convention. Popular conventions include:
1413:
2551:
1099:
1028:
571:
350:
1852:
3055:
1418:
655:
919:
1277:
1582:{\displaystyle {\begin{aligned}\mathbb {P} (y{\mbox{ received}}\mid x{\mbox{ sent}})&{}=(1-p)^{n-d}\cdot p^{d}\\&{}=(1-p)^{n}\cdot \left({\frac {p}{1-p}}\right)^{d}\\\end{aligned}}}
1717:
977:
99:
1958:
1918:
1164:
482:
266:
2340:
2916:
183:
2125:
1326:
1402:
2623:
2778:
2753:
2708:
2384:
2215:
2154:
2066:
1194:
514:
296:
218:
2251:
1990:
2807:
2186:
2040:
149:
2480:
2407:
2979:
2956:
2936:
2867:
2847:
2827:
2728:
2683:
2663:
2643:
2597:
2457:
2427:
2010:
1875:
1800:
1780:
1757:
1737:
1635:
1346:
1320:
1300:
1048:
939:
638:
615:
595:
393:
373:
123:
1650:. They may be unreasonable for other media, such as a DVD, where a single scratch on the disk can cause an error in many neighbouring symbols or codewords.
3401:
2576:-probabilistic methods all based on the observation that it is easier to guess enough error-free positions, than it is to guess all the error-positions.
2488:
1053:
982:
525:
304:
1114:
The maximum likelihood decoding algorithm is an instance of the "marginalize a product function" problem which is solved by applying the
17:
1808:
2984:
3214:
3160:
640:
was sent. If all codewords are equally likely to be sent then this scheme is equivalent to ideal observer decoding. In fact, by
3158:
Feldman, Jon; Wainwright, Martin J.; Karger, David R. (March 2005). "Using Linear
Programming to Decode Binary Linear Codes".
3446:
3415:
3387:
3327:
3294:
46:. There have been many common methods of mapping messages to codewords. These are often used to recover messages sent over a
3082:) is a method for converting the weak analog signal from the head of a magnetic disk or tape drive into a digital signal.
1640:
Errors are independent events – an error at one position in the message does not affect other positions.
886:
421:
3257:
1206:
1682:
944:
64:
1877:
errors are made then minimum distance decoding will still correctly decode the incorrectly transmitted codeword).
3133:
1923:
1883:
1129:
447:
231:
3434:
2259:
1115:
154:
3249:
2074:
517:
424:, mark the ambiguous bits of the codeword as erasures and hope that the outer code disambiguates them
39:
3174:
3097:
2350:
1647:
1611:. Minimum distance decoding is a reasonable decoding method when the following conditions are met:
411:
51:
3096:
A Viterbi decoder uses the
Viterbi algorithm for decoding a bitstream that has been encoded using
3379:
2872:
1362:
618:
2958:
contained no errors, and hence equalled the positions of the sent codeword, then we may decode.
3465:
3241:
3169:
2602:
3206:
2356:
2194:
2133:
2045:
1173:
493:
275:
188:
3407:
2224:
1963:
2783:
2162:
8:
3128:
2573:
2015:
1760:
1108:
1104:
As with ideal observer decoding, a convention must be agreed to for non-unique decoding.
417:
Choose any random codeword from the set of most likely codewords which is nearer to that.
128:
2758:
2733:
2688:
2462:
2389:
1653:
As with other decoding methods, a convention must be agreed to for non-unique decoding.
3427:
3372:
3333:
3187:
3108:
2964:
2941:
2921:
2852:
2832:
2812:
2713:
2668:
2648:
2628:
2582:
2442:
2412:
1995:
1860:
1785:
1765:
1742:
1722:
1620:
1331:
1305:
1285:
1033:
924:
623:
600:
580:
485:
439:
378:
358:
108:
2981:
errors occurred, the probability of such a fortunate selection of columns is given by
3442:
3411:
3383:
3323:
3290:
3253:
3337:
3315:
3282:
3223:
3191:
3179:
3101:
1197:
3438:
3278:
3138:
3091:
2433:
2156:. Syndrome decoding takes advantage of the property of the parity matrix that:
1608:
3459:
3061:
2562:
641:
47:
31:
3183:
2546:{\displaystyle {\begin{matrix}\sum _{i=0}^{t}{\binom {n}{i}}\\\end{matrix}}}
1676:
using a reduced lookup table. This is allowed by the linearity of the code.
3319:
3119:
Optimal decision decoding algorithm (ODDA) for an asymmetric TWRC system.
2432:
Note that this is already of significantly less complexity than that of a
3353:
Optimal decision decoding algorithm (ODDA) for an asymmetric TWRC system;
3273:
Stern, Jacques (1989). "A method for finding codewords of small weight".
2459:
errors were made during transmission, the receiver can look up the value
1665:
102:
1992:
is received. Ordinary minimum distance decoding would lookup the vector
979:
is constant as all codewords are equally likely to be sent. Therefore,
3397:
3314:. Lecture Notes in Computer Science. Vol. 1514. pp. 187–199.
3286:
3227:
1672:, i.e. one on which errors are made. In essence, syndrome decoding is
1094:{\displaystyle \mathbb {P} (y{\mbox{ sent}}\mid x{\mbox{ received}})}
1023:{\displaystyle \mathbb {P} (x{\mbox{ received}}\mid y{\mbox{ sent}})}
566:{\displaystyle \mathbb {P} (x{\mbox{ received}}\mid y{\mbox{ sent}})}
345:{\displaystyle \mathbb {P} (y{\mbox{ sent}}\mid x{\mbox{ received}})}
2780:
be the sub-vector for the corresponding positions of any codeword
1637:
that an error occurs is independent of the position of the symbol.
1107:
The maximum likelihood decoding problem can also be modeled as an
3060:
This method has been improved in various ways, e.g. by Stern and
2042:
for the nearest match - i.e. an element (not necessarily unique)
3114:
1847:{\displaystyle t=\left\lfloor {\frac {d-1}{2}}\right\rfloor }
1646:
These assumptions may be reasonable for transmissions over a
3355:, Universal Journal of Electrical and Electronic Engineering
3104:
is used as a metric for hard decision
Viterbi decoders. The
3050:{\displaystyle \textstyle {\binom {n-t}{k}}/{\binom {n}{k}}}
3378:. Oxford Applied Mathematics and Computing Science Series.
3234:
3079:
3073:
43:
3157:
3240:
3067:
2988:
2493:
1446:
1433:
1082:
1069:
1011:
998:
960:
902:
851:
828:
801:
788:
752:
729:
716:
683:
670:
554:
541:
410:
Request that the codeword be resent –
333:
320:
3406:. Wiley-Interscience Series in Discrete Mathematics.
2987:
2967:
2944:
2924:
2875:
2855:
2835:
2815:
2786:
2761:
2736:
2716:
2691:
2671:
2651:
2631:
2605:
2585:
2491:
2465:
2445:
2415:
2392:
2359:
2262:
2227:
2197:
2165:
2136:
2077:
2048:
2018:
1998:
1966:
1926:
1886:
1863:
1811:
1788:
1768:
1745:
1725:
1685:
1623:
1416:
1365:
1334:
1308:
1288:
1209:
1176:
1132:
1056:
1036:
985:
947:
927:
889:
653:
626:
603:
583:
528:
496:
450:
381:
361:
307:
278:
234:
191:
157:
131:
111:
67:
38:
is the process of translating received messages into
3403:
Introduction to the theory of error-correcting codes
3277:. Lecture Notes in Computer Science. Vol. 388.
3205:
Aji, Srinivas M.; McEliece, Robert J. (March 2000).
3426:
3371:
3049:
2973:
2950:
2930:
2910:
2861:
2841:
2821:
2801:
2772:
2747:
2722:
2702:
2677:
2657:
2637:
2617:
2591:
2545:
2474:
2451:
2421:
2401:
2378:
2334:
2245:
2209:
2180:
2148:
2119:
2060:
2034:
2004:
1984:
1952:
1912:
1869:
1857:errors made by the channel (since if no more than
1846:
1794:
1774:
1751:
1731:
1711:
1629:
1596:is less than one half) is maximised by minimising
1581:
1396:
1340:
1314:
1294:
1271:
1188:
1158:
1093:
1042:
1022:
971:
933:
913:
872:
632:
609:
589:
565:
508:
476:
387:
375:that is most likely to be received as the message
367:
344:
290:
260:
212:
177:
143:
117:
93:
2353:, one has to look-up a precomputed table of size
914:{\displaystyle \mathbb {P} (x{\mbox{ received}})}
3457:
3198:
3111:is used as a metric for soft decision decoders.
2755:will have full rank, which means that if we let
2439:However, under the assumption that no more than
2346:
3350:
1920:is sent over the channel and the error pattern
1272:{\displaystyle d(x,y)=\#\{i:x_{i}\not =y_{i}\}}
433:
355:For example, a person can choose the codeword
3040:
3027:
3013:
2992:
2533:
2520:
1712:{\displaystyle C\subset \mathbb {F} _{2}^{n}}
1607:. It can be assisted or automated by using a
972:{\displaystyle \mathbb {P} (y{\mbox{ sent}})}
94:{\displaystyle C\subset \mathbb {F} _{2}^{n}}
1266:
1234:
1121:
3204:
3151:
2567:
1664:is a highly efficient method of decoding a
1603:Minimum distance decoding is also known as
1325:Note that if the probability of error on a
1030:is maximised as a function of the variable
3115:Optimal decision decoding algorithm (ODDA)
223:
3266:
3173:
1953:{\displaystyle e\in \mathbb {F} _{2}^{n}}
1935:
1913:{\displaystyle x\in \mathbb {F} _{2}^{n}}
1895:
1694:
1422:
1393:
1159:{\displaystyle x\in \mathbb {F} _{2}^{n}}
1141:
1058:
987:
949:
891:
840:
817:
777:
741:
705:
659:
530:
477:{\displaystyle x\in \mathbb {F} _{2}^{n}}
459:
309:
261:{\displaystyle x\in \mathbb {F} _{2}^{n}}
243:
160:
76:
3424:
3303:
2579:The simplest form is due to Prange: Let
298:. The process results in this solution:
220:is the distance between those elements.
3310:Ohta, Kazuo; Pei, Dingyi, eds. (1998).
3309:
3215:IEEE Transactions on Information Theory
3161:IEEE Transactions on Information Theory
2335:{\displaystyle Hz=H(x+e)=Hx+He=0+He=He}
427:Report a decoding failure to the system
398:
14:
3458:
3396:
3312:Advances in Cryptology — ASIACRYPT'98
3272:
3078:Partial response maximum likelihood (
2918:. Hence, if we were lucky that these
1348:is strictly less than one half, then
1101:is maximised, and the claim follows.
3369:
1656:
597:that maximizes the probability that
178:{\displaystyle \mathbb {F} _{2}^{n}}
3100:based on a convolutional code. The
3068:Partial response maximum likelihood
2482:in a further reduced table of size
24:
3437:(GTM). Vol. 86 (2 ed.).
3363:
3207:"The Generalized Distributive Law"
3085:
3031:
2996:
2524:
1231:
25:
3477:
2120:{\displaystyle d(c,z)\leq d(y,z)}
2556:
1302:that is as close as possible to
3374:A first course in coding theory
2938:positions of the received word
2710:the corresponding submatrix of
1802:is capable of correcting up to
3344:
3275:Coding Theory and Applications
3134:Error detection and correction
2730:. With reasonable probability
2287:
2275:
2114:
2102:
2093:
2081:
2028:
2020:
1530:
1517:
1477:
1464:
1452:
1426:
1381:
1369:
1225:
1213:
1088:
1062:
1017:
991:
966:
953:
908:
895:
857:
844:
834:
821:
807:
781:
758:
745:
735:
709:
689:
663:
560:
534:
339:
313:
207:
195:
13:
1:
3435:Graduate Texts in Mathematics
3429:Introduction to Coding Theory
3425:van Lint, Jacobus H. (1992).
3144:
228:One may be given the message
27:Algorithms to decode messages
1880:Now suppose that a codeword
1116:generalized distributive law
7:
3122:
2911:{\displaystyle m=c'G'^{-1}}
1719:is a linear code of length
1397:{\displaystyle d(x,y)=d,\,}
1354:maximum likelihood decoding
1327:discrete memoryless channel
434:Maximum likelihood decoding
57:
10:
3482:
3250:Cambridge University Press
3089:
3071:
2645:used for encoding. Select
2560:
1605:nearest neighbour decoding
1126:Given a received codeword
437:
3244:; Rosenbaum, Ute (1998).
2685:at random, and denote by
2618:{\displaystyle k\times n}
1674:minimum distance decoding
1350:minimum distance decoding
1282:i.e. choose the codeword
1168:minimum distance decoding
1122:Minimum distance decoding
18:Minimum distance decoding
3351:Siamack Ghadimi (2020),
3098:forward error correction
2568:Information set decoding
2351:binary symmetric channel
1648:binary symmetric channel
444:Given a received vector
412:automatic repeat-request
52:binary symmetric channel
3380:Oxford University Press
3242:Beutelspacher, Albrecht
3184:10.1109/TIT.2004.842696
2434:standard array decoding
2379:{\displaystyle 2^{n-k}}
272:generates the codeword
270:ideal observer decoding
224:Ideal observer decoding
3370:Hill, Raymond (1986).
3051:
2975:
2952:
2932:
2912:
2863:
2843:
2823:
2803:
2774:
2749:
2724:
2704:
2679:
2659:
2639:
2619:
2593:
2547:
2516:
2476:
2453:
2423:
2403:
2380:
2336:
2247:
2211:
2210:{\displaystyle x\in C}
2182:
2150:
2149:{\displaystyle y\in C}
2121:
2062:
2061:{\displaystyle c\in C}
2036:
2006:
1986:
1954:
1914:
1871:
1848:
1796:
1776:
1753:
1733:
1713:
1631:
1583:
1398:
1342:
1316:
1296:
1273:
1190:
1189:{\displaystyle y\in C}
1160:
1095:
1044:
1024:
973:
935:
915:
874:
634:
611:
591:
577:that is, the codeword
567:
510:
509:{\displaystyle y\in C}
478:
389:
369:
346:
292:
291:{\displaystyle y\in C}
262:
214:
213:{\displaystyle d(x,y)}
179:
145:
119:
95:
3408:John Wiley & Sons
3320:10.1007/3-540-49649-1
3052:
2976:
2953:
2933:
2913:
2864:
2844:
2824:
2804:
2775:
2750:
2725:
2705:
2680:
2660:
2640:
2620:
2594:
2548:
2496:
2477:
2454:
2424:
2404:
2381:
2337:
2248:
2246:{\displaystyle z=x+e}
2212:
2183:
2151:
2122:
2063:
2037:
2007:
1987:
1985:{\displaystyle z=x+e}
1955:
1915:
1872:
1849:
1797:
1777:
1754:
1739:and minimum distance
1734:
1714:
1632:
1584:
1399:
1343:
1317:
1297:
1274:
1191:
1161:
1096:
1045:
1025:
974:
936:
916:
875:
635:
612:
592:
568:
511:
479:
438:Further information:
390:
370:
347:
293:
263:
215:
180:
151:shall be elements of
146:
120:
96:
3281:. pp. 106–113.
2985:
2965:
2942:
2922:
2873:
2853:
2833:
2813:
2802:{\displaystyle c=mG}
2784:
2759:
2734:
2714:
2689:
2669:
2649:
2629:
2625:generator matrix of
2603:
2583:
2572:This is a family of
2489:
2463:
2443:
2413:
2390:
2357:
2260:
2225:
2195:
2181:{\displaystyle Hx=0}
2163:
2134:
2075:
2046:
2016:
1996:
1964:
1924:
1884:
1861:
1809:
1786:
1766:
1743:
1723:
1683:
1621:
1414:
1363:
1332:
1306:
1286:
1207:
1174:
1130:
1054:
1034:
983:
945:
941:is restructured and
925:
887:
651:
624:
601:
581:
526:
494:
448:
422:another code follows
399:Decoding conventions
395:after transmission.
379:
359:
305:
276:
232:
189:
155:
129:
109:
65:
3246:Projective Geometry
2035:{\displaystyle |C|}
2012:in a table of size
1949:
1909:
1761:parity-check matrix
1708:
1155:
1109:integer programming
473:
257:
174:
144:{\displaystyle x,y}
90:
3287:10.1007/BFb0019850
3109:Euclidean distance
3047:
3046:
2971:
2948:
2928:
2908:
2859:
2839:
2819:
2799:
2773:{\displaystyle c'}
2770:
2748:{\displaystyle G'}
2745:
2720:
2703:{\displaystyle G'}
2700:
2675:
2655:
2635:
2615:
2589:
2543:
2541:
2475:{\displaystyle He}
2472:
2449:
2419:
2402:{\displaystyle He}
2399:
2376:
2332:
2253:is defined to be:
2243:
2207:
2178:
2146:
2117:
2058:
2032:
2002:
1982:
1950:
1933:
1910:
1893:
1867:
1844:
1792:
1772:
1749:
1729:
1709:
1692:
1627:
1579:
1577:
1450:
1437:
1394:
1338:
1312:
1292:
1269:
1186:
1156:
1139:
1091:
1086:
1073:
1040:
1020:
1015:
1002:
969:
964:
931:
911:
906:
870:
868:
855:
832:
805:
792:
756:
733:
720:
687:
674:
630:
607:
587:
563:
558:
545:
506:
486:maximum likelihood
474:
457:
440:Maximum likelihood
385:
365:
342:
337:
324:
288:
258:
241:
210:
175:
158:
141:
115:
91:
74:
3448:978-3-540-54894-2
3417:978-0-471-08684-0
3389:978-0-19-853803-5
3329:978-3-540-65109-3
3296:978-3-540-51643-9
3228:10.1109/18.825794
3038:
3011:
2974:{\displaystyle t}
2951:{\displaystyle y}
2931:{\displaystyle k}
2862:{\displaystyle m}
2849:, we can recover
2842:{\displaystyle m}
2822:{\displaystyle C}
2723:{\displaystyle G}
2678:{\displaystyle G}
2658:{\displaystyle k}
2638:{\displaystyle C}
2592:{\displaystyle G}
2531:
2452:{\displaystyle t}
2422:{\displaystyle e}
2005:{\displaystyle z}
1870:{\displaystyle t}
1838:
1795:{\displaystyle C}
1775:{\displaystyle H}
1752:{\displaystyle d}
1732:{\displaystyle n}
1662:Syndrome decoding
1657:Syndrome decoding
1630:{\displaystyle p}
1563:
1449:
1436:
1352:is equivalent to
1341:{\displaystyle p}
1315:{\displaystyle x}
1295:{\displaystyle y}
1170:picks a codeword
1085:
1072:
1043:{\displaystyle y}
1014:
1001:
963:
934:{\displaystyle x}
905:
861:
854:
831:
804:
791:
762:
755:
732:
719:
686:
673:
633:{\displaystyle y}
610:{\displaystyle x}
590:{\displaystyle y}
557:
544:
490:picks a codeword
388:{\displaystyle x}
368:{\displaystyle y}
336:
323:
118:{\displaystyle n}
16:(Redirected from
3473:
3452:
3432:
3421:
3393:
3377:
3357:
3356:
3348:
3342:
3341:
3307:
3301:
3300:
3270:
3264:
3263:
3238:
3232:
3231:
3211:
3202:
3196:
3195:
3177:
3155:
3129:Don't care alarm
3102:Hamming distance
3056:
3054:
3053:
3048:
3045:
3044:
3043:
3030:
3023:
3018:
3017:
3016:
3007:
2995:
2980:
2978:
2977:
2972:
2957:
2955:
2954:
2949:
2937:
2935:
2934:
2929:
2917:
2915:
2914:
2909:
2907:
2906:
2905:
2889:
2868:
2866:
2865:
2860:
2848:
2846:
2845:
2840:
2828:
2826:
2825:
2820:
2808:
2806:
2805:
2800:
2779:
2777:
2776:
2771:
2769:
2754:
2752:
2751:
2746:
2744:
2729:
2727:
2726:
2721:
2709:
2707:
2706:
2701:
2699:
2684:
2682:
2681:
2676:
2664:
2662:
2661:
2656:
2644:
2642:
2641:
2636:
2624:
2622:
2621:
2616:
2598:
2596:
2595:
2590:
2552:
2550:
2549:
2544:
2542:
2538:
2537:
2536:
2523:
2515:
2510:
2481:
2479:
2478:
2473:
2458:
2456:
2455:
2450:
2428:
2426:
2425:
2420:
2408:
2406:
2405:
2400:
2385:
2383:
2382:
2377:
2375:
2374:
2341:
2339:
2338:
2333:
2252:
2250:
2249:
2244:
2221:of the received
2216:
2214:
2213:
2208:
2187:
2185:
2184:
2179:
2155:
2153:
2152:
2147:
2126:
2124:
2123:
2118:
2067:
2065:
2064:
2059:
2041:
2039:
2038:
2033:
2031:
2023:
2011:
2009:
2008:
2003:
1991:
1989:
1988:
1983:
1959:
1957:
1956:
1951:
1948:
1943:
1938:
1919:
1917:
1916:
1911:
1908:
1903:
1898:
1876:
1874:
1873:
1868:
1853:
1851:
1850:
1845:
1843:
1839:
1834:
1823:
1801:
1799:
1798:
1793:
1781:
1779:
1778:
1773:
1758:
1756:
1755:
1750:
1738:
1736:
1735:
1730:
1718:
1716:
1715:
1710:
1707:
1702:
1697:
1636:
1634:
1633:
1628:
1617:The probability
1588:
1586:
1585:
1580:
1578:
1574:
1573:
1568:
1564:
1562:
1548:
1538:
1537:
1513:
1508:
1504:
1503:
1491:
1490:
1460:
1451:
1447:
1438:
1434:
1425:
1403:
1401:
1400:
1395:
1347:
1345:
1344:
1339:
1321:
1319:
1318:
1313:
1301:
1299:
1298:
1293:
1278:
1276:
1275:
1270:
1265:
1264:
1252:
1251:
1198:Hamming distance
1196:to minimise the
1195:
1193:
1192:
1187:
1165:
1163:
1162:
1157:
1154:
1149:
1144:
1100:
1098:
1097:
1092:
1087:
1083:
1074:
1070:
1061:
1049:
1047:
1046:
1041:
1029:
1027:
1026:
1021:
1016:
1012:
1003:
999:
990:
978:
976:
975:
970:
965:
961:
952:
940:
938:
937:
932:
920:
918:
917:
912:
907:
903:
894:
879:
877:
876:
871:
869:
862:
860:
856:
852:
843:
837:
833:
829:
820:
814:
806:
802:
793:
789:
780:
772:
767:
763:
761:
757:
753:
744:
738:
734:
730:
721:
717:
708:
702:
697:
688:
684:
675:
671:
662:
639:
637:
636:
631:
616:
614:
613:
608:
596:
594:
593:
588:
572:
570:
569:
564:
559:
555:
546:
542:
533:
515:
513:
512:
507:
483:
481:
480:
475:
472:
467:
462:
394:
392:
391:
386:
374:
372:
371:
366:
351:
349:
348:
343:
338:
334:
325:
321:
312:
297:
295:
294:
289:
267:
265:
264:
259:
256:
251:
246:
219:
217:
216:
211:
184:
182:
181:
176:
173:
168:
163:
150:
148:
147:
142:
124:
122:
121:
116:
105:with the length
101:is considered a
100:
98:
97:
92:
89:
84:
79:
21:
3481:
3480:
3476:
3475:
3474:
3472:
3471:
3470:
3456:
3455:
3449:
3439:Springer-Verlag
3418:
3390:
3366:
3364:Further reading
3361:
3360:
3349:
3345:
3330:
3308:
3304:
3297:
3279:Springer-Verlag
3271:
3267:
3260:
3252:. p. 190.
3239:
3235:
3209:
3203:
3199:
3175:10.1.1.111.6585
3156:
3152:
3147:
3139:Forbidden input
3125:
3117:
3094:
3092:Viterbi decoder
3088:
3086:Viterbi decoder
3076:
3070:
3039:
3026:
3025:
3024:
3019:
3012:
2997:
2991:
2990:
2989:
2986:
2983:
2982:
2966:
2963:
2962:
2943:
2940:
2939:
2923:
2920:
2919:
2898:
2894:
2890:
2882:
2874:
2871:
2870:
2854:
2851:
2850:
2834:
2831:
2830:
2814:
2811:
2810:
2785:
2782:
2781:
2762:
2760:
2757:
2756:
2737:
2735:
2732:
2731:
2715:
2712:
2711:
2692:
2690:
2687:
2686:
2670:
2667:
2666:
2650:
2647:
2646:
2630:
2627:
2626:
2604:
2601:
2600:
2584:
2581:
2580:
2570:
2565:
2559:
2540:
2539:
2532:
2519:
2518:
2517:
2511:
2500:
2492:
2490:
2487:
2486:
2464:
2461:
2460:
2444:
2441:
2440:
2414:
2411:
2410:
2391:
2388:
2387:
2364:
2360:
2358:
2355:
2354:
2261:
2258:
2257:
2226:
2223:
2222:
2196:
2193:
2192:
2164:
2161:
2160:
2135:
2132:
2131:
2076:
2073:
2072:
2047:
2044:
2043:
2027:
2019:
2017:
2014:
2013:
1997:
1994:
1993:
1965:
1962:
1961:
1944:
1939:
1934:
1925:
1922:
1921:
1904:
1899:
1894:
1885:
1882:
1881:
1862:
1859:
1858:
1824:
1822:
1818:
1810:
1807:
1806:
1787:
1784:
1783:
1782:. Then clearly
1767:
1764:
1763:
1744:
1741:
1740:
1724:
1721:
1720:
1703:
1698:
1693:
1684:
1681:
1680:
1659:
1622:
1619:
1618:
1576:
1575:
1569:
1552:
1547:
1543:
1542:
1533:
1529:
1512:
1506:
1505:
1499:
1495:
1480:
1476:
1459:
1455:
1445:
1432:
1421:
1417:
1415:
1412:
1411:
1364:
1361:
1360:
1333:
1330:
1329:
1307:
1304:
1303:
1287:
1284:
1283:
1260:
1256:
1247:
1243:
1208:
1205:
1204:
1175:
1172:
1171:
1150:
1145:
1140:
1131:
1128:
1127:
1124:
1081:
1068:
1057:
1055:
1052:
1051:
1050:precisely when
1035:
1032:
1031:
1010:
997:
986:
984:
981:
980:
959:
948:
946:
943:
942:
926:
923:
922:
901:
890:
888:
885:
884:
867:
866:
850:
839:
838:
827:
816:
815:
813:
800:
787:
776:
771:
765:
764:
751:
740:
739:
728:
715:
704:
703:
701:
696:
692:
682:
669:
658:
654:
652:
649:
648:
625:
622:
621:
602:
599:
598:
582:
579:
578:
553:
540:
529:
527:
524:
523:
495:
492:
491:
468:
463:
458:
449:
446:
445:
442:
436:
401:
380:
377:
376:
360:
357:
356:
332:
319:
308:
306:
303:
302:
277:
274:
273:
252:
247:
242:
233:
230:
229:
226:
190:
187:
186:
169:
164:
159:
156:
153:
152:
130:
127:
126:
110:
107:
106:
85:
80:
75:
66:
63:
62:
60:
28:
23:
22:
15:
12:
11:
5:
3479:
3469:
3468:
3454:
3453:
3447:
3422:
3416:
3394:
3388:
3365:
3362:
3359:
3358:
3343:
3328:
3302:
3295:
3265:
3258:
3233:
3222:(2): 325–343.
3197:
3168:(3): 954–972.
3149:
3148:
3146:
3143:
3142:
3141:
3136:
3131:
3124:
3121:
3116:
3113:
3090:Main article:
3087:
3084:
3072:Main article:
3069:
3066:
3064:and Sendrier.
3042:
3037:
3034:
3029:
3022:
3015:
3010:
3006:
3003:
3000:
2994:
2970:
2947:
2927:
2904:
2901:
2897:
2893:
2888:
2885:
2881:
2878:
2858:
2838:
2829:for a message
2818:
2798:
2795:
2792:
2789:
2768:
2765:
2743:
2740:
2719:
2698:
2695:
2674:
2654:
2634:
2614:
2611:
2608:
2588:
2569:
2566:
2561:Main article:
2558:
2555:
2554:
2553:
2535:
2530:
2527:
2522:
2514:
2509:
2506:
2503:
2499:
2495:
2494:
2471:
2468:
2448:
2418:
2398:
2395:
2373:
2370:
2367:
2363:
2343:
2342:
2331:
2328:
2325:
2322:
2319:
2316:
2313:
2310:
2307:
2304:
2301:
2298:
2295:
2292:
2289:
2286:
2283:
2280:
2277:
2274:
2271:
2268:
2265:
2242:
2239:
2236:
2233:
2230:
2206:
2203:
2200:
2189:
2188:
2177:
2174:
2171:
2168:
2145:
2142:
2139:
2128:
2127:
2116:
2113:
2110:
2107:
2104:
2101:
2098:
2095:
2092:
2089:
2086:
2083:
2080:
2057:
2054:
2051:
2030:
2026:
2022:
2001:
1981:
1978:
1975:
1972:
1969:
1947:
1942:
1937:
1932:
1929:
1907:
1902:
1897:
1892:
1889:
1866:
1855:
1854:
1842:
1837:
1833:
1830:
1827:
1821:
1817:
1814:
1791:
1771:
1748:
1728:
1706:
1701:
1696:
1691:
1688:
1658:
1655:
1644:
1643:
1642:
1641:
1638:
1626:
1609:standard array
1590:
1589:
1572:
1567:
1561:
1558:
1555:
1551:
1546:
1541:
1536:
1532:
1528:
1525:
1522:
1519:
1516:
1511:
1509:
1507:
1502:
1498:
1494:
1489:
1486:
1483:
1479:
1475:
1472:
1469:
1466:
1463:
1458:
1456:
1454:
1444:
1441:
1435: received
1431:
1428:
1424:
1420:
1419:
1405:
1404:
1392:
1389:
1386:
1383:
1380:
1377:
1374:
1371:
1368:
1337:
1311:
1291:
1280:
1279:
1268:
1263:
1259:
1255:
1250:
1246:
1242:
1239:
1236:
1233:
1230:
1227:
1224:
1221:
1218:
1215:
1212:
1185:
1182:
1179:
1153:
1148:
1143:
1138:
1135:
1123:
1120:
1090:
1084: received
1080:
1077:
1067:
1064:
1060:
1039:
1019:
1009:
1006:
1000: received
996:
993:
989:
968:
958:
955:
951:
930:
910:
904: received
900:
897:
893:
881:
880:
865:
859:
849:
846:
842:
836:
830: received
826:
823:
819:
812:
809:
803: received
799:
796:
786:
783:
779:
775:
770:
768:
766:
760:
750:
747:
743:
737:
727:
724:
718: received
714:
711:
707:
700:
695:
693:
691:
681:
678:
672: received
668:
665:
661:
657:
656:
629:
617:was received,
606:
586:
575:
574:
562:
552:
549:
543: received
539:
536:
532:
505:
502:
499:
471:
466:
461:
456:
453:
435:
432:
431:
430:
429:
428:
425:
418:
415:
400:
397:
384:
364:
353:
352:
341:
335: received
331:
328:
318:
315:
311:
287:
284:
281:
255:
250:
245:
240:
237:
225:
222:
209:
206:
203:
200:
197:
194:
172:
167:
162:
140:
137:
134:
114:
88:
83:
78:
73:
70:
59:
56:
26:
9:
6:
4:
3:
2:
3478:
3467:
3466:Coding theory
3464:
3463:
3461:
3450:
3444:
3440:
3436:
3431:
3430:
3423:
3419:
3413:
3409:
3405:
3404:
3399:
3395:
3391:
3385:
3381:
3376:
3375:
3368:
3367:
3354:
3347:
3339:
3335:
3331:
3325:
3321:
3317:
3313:
3306:
3298:
3292:
3288:
3284:
3280:
3276:
3269:
3261:
3259:0-521-48277-1
3255:
3251:
3247:
3243:
3237:
3229:
3225:
3221:
3217:
3216:
3208:
3201:
3193:
3189:
3185:
3181:
3176:
3171:
3167:
3163:
3162:
3154:
3150:
3140:
3137:
3135:
3132:
3130:
3127:
3126:
3120:
3112:
3110:
3107:
3103:
3099:
3093:
3083:
3081:
3075:
3065:
3063:
3058:
3035:
3032:
3020:
3008:
3004:
3001:
2998:
2968:
2959:
2945:
2925:
2902:
2899:
2895:
2891:
2886:
2883:
2879:
2876:
2856:
2836:
2816:
2796:
2793:
2790:
2787:
2766:
2763:
2741:
2738:
2717:
2696:
2693:
2672:
2652:
2632:
2612:
2609:
2606:
2586:
2577:
2575:
2564:
2563:List decoding
2557:List decoding
2528:
2525:
2512:
2507:
2504:
2501:
2497:
2485:
2484:
2483:
2469:
2466:
2446:
2437:
2435:
2430:
2416:
2396:
2393:
2371:
2368:
2365:
2361:
2352:
2348:
2329:
2326:
2323:
2320:
2317:
2314:
2311:
2308:
2305:
2302:
2299:
2296:
2293:
2290:
2284:
2281:
2278:
2272:
2269:
2266:
2263:
2256:
2255:
2254:
2240:
2237:
2234:
2231:
2228:
2220:
2204:
2201:
2198:
2175:
2172:
2169:
2166:
2159:
2158:
2157:
2143:
2140:
2137:
2111:
2108:
2105:
2099:
2096:
2090:
2087:
2084:
2078:
2071:
2070:
2069:
2055:
2052:
2049:
2024:
1999:
1979:
1976:
1973:
1970:
1967:
1960:occurs. Then
1945:
1940:
1930:
1927:
1905:
1900:
1890:
1887:
1878:
1864:
1840:
1835:
1831:
1828:
1825:
1819:
1815:
1812:
1805:
1804:
1803:
1789:
1769:
1762:
1746:
1726:
1704:
1699:
1689:
1686:
1679:Suppose that
1677:
1675:
1671:
1670:noisy channel
1667:
1663:
1654:
1651:
1649:
1639:
1624:
1616:
1615:
1614:
1613:
1612:
1610:
1606:
1601:
1599:
1595:
1592:which (since
1570:
1565:
1559:
1556:
1553:
1549:
1544:
1539:
1534:
1526:
1523:
1520:
1514:
1510:
1500:
1496:
1492:
1487:
1484:
1481:
1473:
1470:
1467:
1461:
1457:
1442:
1439:
1429:
1410:
1409:
1408:
1390:
1387:
1384:
1378:
1375:
1372:
1366:
1359:
1358:
1357:
1355:
1351:
1335:
1328:
1323:
1309:
1289:
1261:
1257:
1253:
1248:
1244:
1240:
1237:
1228:
1222:
1219:
1216:
1210:
1203:
1202:
1201:
1199:
1183:
1180:
1177:
1169:
1151:
1146:
1136:
1133:
1119:
1117:
1112:
1110:
1105:
1102:
1078:
1075:
1065:
1037:
1007:
1004:
994:
956:
928:
898:
863:
847:
824:
810:
797:
794:
784:
773:
769:
748:
725:
722:
712:
698:
694:
679:
676:
666:
647:
646:
645:
643:
642:Bayes Theorem
627:
620:
604:
584:
550:
547:
537:
522:
521:
520:
519:
503:
500:
497:
489:
487:
469:
464:
454:
451:
441:
426:
423:
419:
416:
413:
409:
408:
407:
406:
405:
396:
382:
362:
329:
326:
316:
301:
300:
299:
285:
282:
279:
271:
253:
248:
238:
235:
221:
204:
201:
198:
192:
170:
165:
138:
135:
132:
112:
104:
86:
81:
71:
68:
55:
53:
49:
48:noisy channel
45:
41:
37:
33:
32:coding theory
19:
3428:
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354:
269:
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61:
50:, such as a
35:
29:
3398:Pless, Vera
2665:columns of
2347:ML decoding
2345:To perform
1666:linear code
1356:, since if
103:binary code
42:of a given
3145:References
2386:, mapping
1448: sent
1071: sent
1013: sent
962: sent
853: sent
790: sent
754: sent
731: sent
685: sent
619:given that
556: sent
322: sent
3170:CiteSeerX
3002:−
2900:−
2610:×
2574:Las Vegas
2498:∑
2369:−
2202:∈
2141:∈
2097:≤
2053:∈
1931:∈
1891:∈
1829:−
1690:⊂
1557:−
1540:⋅
1524:−
1493:⋅
1485:−
1471:−
1440:∣
1232:#
1181:∈
1137:∈
1111:problem.
1076:∣
1005:∣
811:⋅
795:∣
677:∣
548:∣
518:maximizes
501:∈
455:∈
327:∣
283:∈
239:∈
72:⊂
40:codewords
3460:Category
3400:(1982).
3338:37257901
3123:See also
3062:Canteaut
2896:′
2887:′
2767:′
2742:′
2697:′
2219:syndrome
2191:for all
2130:for all
1841:⌋
1820:⌊
1254:≠
488:decoding
58:Notation
36:decoding
3192:3120399
3106:squared
2599:be the
1668:over a
268:, then
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185:; and
3334:S2CID
3210:(PDF)
3188:S2CID
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2068:with
1759:with
516:that
3443:ISBN
3412:ISBN
3384:ISBN
3324:ISBN
3291:ISBN
3254:ISBN
3080:PRML
3074:PRML
44:code
3316:doi
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