1672:
of the quadratic residue codes with the code length up to 113. However, decoding of long binary quadratic residue codes and non-binary quadratic residue codes continue to be a challenge. Currently, decoding quadratic residue codes is still an active research area in the theory of error-correcting
1626:
Since late 1980, there are many algebraic decoding algorithms were developed for correcting errors on quadratic residue codes. These algorithms can achieve the (true) error-correcting capacity
475:
1376:
949:
1670:
1148:
568:
1461:
1577:
990:
839:
1616:
1503:
1737:
Y. Li, Y. Duan, H. C. Chang, H. Liu, T. K. Truong, Using the difference of syndromes to decode quadratic residue codes, IEEE Trans. Inf. Theory 64(7), 5179-5190 (2018)
1181:
772:
136:
1071:
792:
588:
203:
732:
640:
298:
238:
171:
104:
69:
1405:
1313:
1016:
1204:
1523:
1437:
1284:
1264:
1244:
1224:
1091:
1036:
879:
859:
812:
700:
680:
660:
608:
518:
498:
402:
378:
358:
338:
318:
266:
1719:
Reed, I.S., Truong, T.K., Chen, X., Yin, X., The algebraic decoding of the (41, 21, 9) quadratic residue code. IEEE Trans. Inf. Theory 38(3), 974–986 (1992)
1713:
M. Elia, Algebraic decoding of the (23,12,7) Golay code, IEEE Transactions on
Information Theory, Volume: 33, Issue: 1, pg. 150-151, January 1987.
1716:
Reed, I.S., Yin, X., Truong, T.K., Algebraic decoding of the (32, 16, 8) quadratic residue code. IEEE Trans. Inf. Theory 36(4), 876–880 (1990)
1731:
He, R., Reed, I.S., Truong, T.K., Chen, X., Decoding the (47, 24, 11) quadratic residue code. IEEE Trans. Inf. Theory 47(3), 1181–1186 (2001)
1725:
Chen, X., Reed, I.S., Truong, T.K., Decoding the (73, 37, 13) quadratic-residue code. IEE Proc., Comput. Digit. Tech. 141(5), 253–258 (1994)
1722:
Humphreys, J.F. Algebraic decoding of the ternary (13, 7, 5) quadratic-residue code. IEEE Trans. Inf. Theory 38(3), 1122–1125 (May 1992)
1728:
Higgs, R.J., Humphreys, J.F.: Decoding the ternary (23, 12, 8) quadratic-residue code. IEE Proc., Comm. 142(3), 129–134 (June 1995)
410:
1750:
1318:
1525:) an extended quadratic residue code is self-dual; otherwise it is equivalent but not equal to its dual. By the
890:
1629:
1537:), the automorphism group of an extended quadratic residue code has a subgroup which is isomorphic to either
1096:
1755:
523:
1442:
205:
1540:
957:
817:
1582:
1482:
381:
138:
1153:
737:
109:
1702:
1041:
777:
573:
176:
705:
613:
271:
211:
144:
77:
42:
1381:
1289:
995:
1186:
1530:
1508:
1422:
1416:
1269:
1249:
1229:
1209:
1076:
1021:
864:
844:
797:
685:
665:
645:
593:
503:
483:
387:
363:
343:
323:
303:
251:
1744:
1534:
1690:
841:
either results in the same code or an equivalent code, according to whether or not
71:
28:
1475:
Adding an overall parity-check digit to a quadratic residue code gives an
17:
1706:
1315:
also generates a quadratic residue code; more precisely the ideal of
1686:, North-Holland Publishing Co., Amsterdam-New York-Oxford, 1977.
884:
An alternative construction avoids roots of unity. Define
404:. Its generator polynomial as a cyclic code is given by
1632:
1585:
1543:
1511:
1485:
1445:
1425:
1384:
1321:
1292:
1272:
1252:
1232:
1212:
1189:
1156:
1099:
1079:
1044:
1024:
998:
960:
893:
867:
847:
820:
800:
780:
740:
708:
688:
668:
648:
616:
596:
576:
526:
506:
486:
413:
390:
366:
346:
326:
306:
274:
254:
214:
179:
147:
112:
80:
45:
610:th root of unity in some finite extension field of
1664:
1610:
1571:
1517:
1497:
1455:
1431:
1399:
1370:
1307:
1278:
1258:
1238:
1218:
1198:
1175:
1142:
1085:
1065:
1030:
1010:
984:
943:
873:
853:
833:
806:
786:
766:
726:
694:
674:
654:
634:
602:
582:
562:
512:
492:
470:{\displaystyle f(x)=\prod _{j\in Q}(x-\zeta ^{j})}
469:
396:
372:
352:
332:
312:
292:
260:
232:
197:
165:
130:
98:
63:
1701:(5), Piscataway, NJ, USA: IEEE Press: 1269–1273,
1693:(September 2006), "The Gleason-Prange theorem",
39:Examples of quadratic residue codes include the
1371:{\displaystyle F_{l}/\langle X^{p}-1\rangle }
8:
1659:
1633:
1365:
1346:
557:
527:
248:There is a quadratic residue code of length
1407:corresponds to the quadratic residue code.
944:{\displaystyle g(x)=c+\sum _{j\in Q}x^{j}}
1651:
1631:
1593:
1584:
1554:
1542:
1510:
1484:
1446:
1444:
1424:
1383:
1353:
1341:
1326:
1320:
1291:
1271:
1251:
1231:
1211:
1188:
1161:
1155:
1132:
1121:
1115:
1098:
1078:
1043:
1023:
997:
959:
935:
919:
892:
866:
846:
825:
819:
799:
779:
756:
739:
707:
687:
667:
647:
615:
595:
575:
525:
505:
485:
458:
433:
412:
389:
365:
345:
325:
305:
273:
253:
213:
178:
146:
111:
79:
44:
1682:F. J. MacWilliams and N. J. A. Sloane,
1665:{\displaystyle \lfloor (d-1)/2\rfloor }
1143:{\displaystyle c=(1+{\sqrt {p^{*}}})/2}
1419:of a quadratic residue code of length
7:
1684:The Theory of Error-Correcting Codes
500:is the set of quadratic residues of
563:{\displaystyle \{1,2,\ldots ,p-1\}}
14:
682:ensures that the coefficients of
1477:extended quadratic residue code
734:. The dimension of the code is
1648:
1636:
1605:
1599:
1566:
1560:
1394:
1388:
1338:
1332:
1302:
1296:
1129:
1106:
1054:
1048:
979:
973:
903:
897:
753:
741:
721:
715:
629:
623:
464:
445:
423:
417:
287:
281:
227:
221:
192:
180:
160:
154:
125:
113:
93:
87:
58:
46:
1:
1456:{\displaystyle {\sqrt {p}}}
1772:
1572:{\displaystyle PSL_{2}(p)}
985:{\displaystyle c\in GF(l)}
861:is a quadratic residue of
834:{\displaystyle \zeta ^{r}}
662:is a quadratic residue of
15:
1611:{\displaystyle SL_{2}(p)}
1498:{\displaystyle p\equiv 3}
16:Not to be confused with
1695:IEEE Trans. Inf. Theory
1176:{\displaystyle p^{*}=p}
767:{\displaystyle (p+1)/2}
131:{\displaystyle (23,12)}
1666:
1612:
1573:
1527:Gleason–Prange theorem
1519:
1499:
1457:
1433:
1401:
1372:
1309:
1280:
1260:
1240:
1220:
1200:
1177:
1144:
1087:
1067:
1066:{\displaystyle g(1)=1}
1032:
1012:
986:
945:
875:
855:
835:
808:
788:
787:{\displaystyle \zeta }
768:
728:
696:
676:
656:
636:
604:
584:
583:{\displaystyle \zeta }
564:
514:
494:
471:
398:
374:
354:
334:
314:
294:
268:over the finite field
262:
234:
199:
198:{\displaystyle (11,6)}
167:
132:
100:
65:
25:quadratic residue code
1667:
1613:
1574:
1520:
1500:
1458:
1434:
1402:
1373:
1310:
1281:
1261:
1241:
1221:
1206:according to whether
1201:
1178:
1145:
1088:
1068:
1033:
1013:
987:
946:
876:
856:
836:
809:
794:by another primitive
789:
769:
729:
727:{\displaystyle GF(l)}
697:
677:
657:
642:. The condition that
637:
635:{\displaystyle GF(l)}
605:
585:
565:
515:
495:
472:
399:
375:
355:
335:
315:
295:
293:{\displaystyle GF(l)}
263:
235:
233:{\displaystyle GF(3)}
200:
168:
166:{\displaystyle GF(2)}
133:
101:
99:{\displaystyle GF(2)}
66:
64:{\displaystyle (7,4)}
1630:
1583:
1541:
1509:
1483:
1443:
1423:
1400:{\displaystyle g(x)}
1382:
1319:
1308:{\displaystyle g(x)}
1290:
1270:
1250:
1230:
1210:
1187:
1154:
1097:
1077:
1042:
1022:
996:
958:
891:
865:
845:
818:
798:
778:
738:
706:
686:
666:
646:
614:
594:
574:
524:
504:
484:
411:
388:
364:
344:
324:
304:
272:
252:
212:
177:
145:
110:
78:
43:
1011:{\displaystyle l=2}
1662:
1608:
1569:
1515:
1495:
1453:
1429:
1397:
1368:
1305:
1276:
1256:
1236:
1216:
1199:{\displaystyle -p}
1196:
1173:
1140:
1083:
1063:
1028:
1008:
982:
941:
930:
871:
851:
831:
814:-th root of unity
804:
784:
764:
724:
692:
672:
652:
632:
600:
580:
560:
510:
490:
467:
444:
394:
370:
350:
330:
310:
290:
258:
230:
206:ternary Golay code
195:
163:
128:
96:
61:
1751:Quadratic residue
1707:10.1109/18.133245
1518:{\displaystyle 4}
1465:square root bound
1451:
1432:{\displaystyle p}
1279:{\displaystyle 4}
1259:{\displaystyle 3}
1239:{\displaystyle 1}
1219:{\displaystyle p}
1127:
1086:{\displaystyle l}
1031:{\displaystyle c}
915:
874:{\displaystyle p}
854:{\displaystyle r}
807:{\displaystyle p}
695:{\displaystyle f}
675:{\displaystyle p}
655:{\displaystyle l}
603:{\displaystyle p}
513:{\displaystyle p}
493:{\displaystyle Q}
429:
397:{\displaystyle p}
382:quadratic residue
373:{\displaystyle l}
353:{\displaystyle p}
333:{\displaystyle l}
313:{\displaystyle p}
261:{\displaystyle p}
139:binary Golay code
1763:
1709:
1671:
1669:
1668:
1663:
1655:
1617:
1615:
1614:
1609:
1598:
1597:
1578:
1576:
1575:
1570:
1559:
1558:
1524:
1522:
1521:
1516:
1504:
1502:
1501:
1496:
1462:
1460:
1459:
1454:
1452:
1447:
1439:is greater than
1438:
1436:
1435:
1430:
1406:
1404:
1403:
1398:
1377:
1375:
1374:
1369:
1358:
1357:
1345:
1331:
1330:
1314:
1312:
1311:
1306:
1285:
1283:
1282:
1277:
1265:
1263:
1262:
1257:
1245:
1243:
1242:
1237:
1226:is congruent to
1225:
1223:
1222:
1217:
1205:
1203:
1202:
1197:
1182:
1180:
1179:
1174:
1166:
1165:
1149:
1147:
1146:
1141:
1136:
1128:
1126:
1125:
1116:
1093:is odd, choose
1092:
1090:
1089:
1084:
1072:
1070:
1069:
1064:
1037:
1035:
1034:
1029:
1017:
1015:
1014:
1009:
991:
989:
988:
983:
950:
948:
947:
942:
940:
939:
929:
880:
878:
877:
872:
860:
858:
857:
852:
840:
838:
837:
832:
830:
829:
813:
811:
810:
805:
793:
791:
790:
785:
773:
771:
770:
765:
760:
733:
731:
730:
725:
701:
699:
698:
693:
681:
679:
678:
673:
661:
659:
658:
653:
641:
639:
638:
633:
609:
607:
606:
601:
589:
587:
586:
581:
569:
567:
566:
561:
519:
517:
516:
511:
499:
497:
496:
491:
476:
474:
473:
468:
463:
462:
443:
403:
401:
400:
395:
379:
377:
376:
371:
359:
357:
356:
351:
339:
337:
336:
331:
319:
317:
316:
311:
299:
297:
296:
291:
267:
265:
264:
259:
239:
237:
236:
231:
204:
202:
201:
196:
172:
170:
169:
164:
137:
135:
134:
129:
105:
103:
102:
97:
70:
68:
67:
62:
1771:
1770:
1766:
1765:
1764:
1762:
1761:
1760:
1741:
1740:
1689:
1679:
1628:
1627:
1624:
1622:Decoding Method
1589:
1581:
1580:
1550:
1539:
1538:
1507:
1506:
1481:
1480:
1473:
1441:
1440:
1421:
1420:
1413:
1380:
1379:
1349:
1322:
1317:
1316:
1288:
1287:
1268:
1267:
1248:
1247:
1228:
1227:
1208:
1207:
1185:
1184:
1157:
1152:
1151:
1117:
1095:
1094:
1075:
1074:
1040:
1039:
1038:to ensure that
1020:
1019:
994:
993:
956:
955:
954:for a suitable
931:
889:
888:
863:
862:
843:
842:
821:
816:
815:
796:
795:
776:
775:
736:
735:
704:
703:
684:
683:
664:
663:
644:
643:
612:
611:
592:
591:
590:is a primitive
572:
571:
522:
521:
502:
501:
482:
481:
454:
409:
408:
386:
385:
362:
361:
342:
341:
322:
321:
302:
301:
270:
269:
250:
249:
246:
210:
209:
175:
174:
143:
142:
108:
107:
76:
75:
41:
40:
37:
21:
12:
11:
5:
1769:
1767:
1759:
1758:
1753:
1743:
1742:
1739:
1738:
1735:
1732:
1729:
1726:
1723:
1720:
1717:
1714:
1711:
1687:
1678:
1675:
1661:
1658:
1654:
1650:
1647:
1644:
1641:
1638:
1635:
1623:
1620:
1607:
1604:
1601:
1596:
1592:
1588:
1568:
1565:
1562:
1557:
1553:
1549:
1546:
1531:Andrew Gleason
1514:
1494:
1491:
1488:
1472:
1469:
1463:; this is the
1450:
1428:
1417:minimum weight
1412:
1409:
1396:
1393:
1390:
1387:
1367:
1364:
1361:
1356:
1352:
1348:
1344:
1340:
1337:
1334:
1329:
1325:
1304:
1301:
1298:
1295:
1275:
1255:
1235:
1215:
1195:
1192:
1172:
1169:
1164:
1160:
1139:
1135:
1131:
1124:
1120:
1114:
1111:
1108:
1105:
1102:
1082:
1062:
1059:
1056:
1053:
1050:
1047:
1027:
1007:
1004:
1001:
981:
978:
975:
972:
969:
966:
963:
952:
951:
938:
934:
928:
925:
922:
918:
914:
911:
908:
905:
902:
899:
896:
870:
850:
828:
824:
803:
783:
763:
759:
755:
752:
749:
746:
743:
723:
720:
717:
714:
711:
691:
671:
651:
631:
628:
625:
622:
619:
599:
579:
559:
556:
553:
550:
547:
544:
541:
538:
535:
532:
529:
509:
489:
478:
477:
466:
461:
457:
453:
450:
447:
442:
439:
436:
432:
428:
425:
422:
419:
416:
393:
369:
349:
329:
309:
289:
286:
283:
280:
277:
257:
245:
242:
229:
226:
223:
220:
217:
194:
191:
188:
185:
182:
162:
159:
156:
153:
150:
127:
124:
121:
118:
115:
95:
92:
89:
86:
83:
60:
57:
54:
51:
48:
36:
33:
13:
10:
9:
6:
4:
3:
2:
1768:
1757:
1756:Coding theory
1754:
1752:
1749:
1748:
1746:
1736:
1733:
1730:
1727:
1724:
1721:
1718:
1715:
1712:
1708:
1704:
1700:
1696:
1692:
1691:Blahut, R. E.
1688:
1685:
1681:
1680:
1676:
1674:
1656:
1652:
1645:
1642:
1639:
1621:
1619:
1602:
1594:
1590:
1586:
1563:
1555:
1551:
1547:
1544:
1536:
1535:Eugene Prange
1532:
1528:
1512:
1492:
1489:
1486:
1478:
1471:Extended code
1470:
1468:
1466:
1448:
1426:
1418:
1410:
1408:
1391:
1385:
1378:generated by
1362:
1359:
1354:
1350:
1342:
1335:
1327:
1323:
1299:
1293:
1273:
1253:
1233:
1213:
1193:
1190:
1170:
1167:
1162:
1158:
1137:
1133:
1122:
1118:
1112:
1109:
1103:
1100:
1080:
1060:
1057:
1051:
1045:
1025:
1005:
1002:
999:
976:
970:
967:
964:
961:
936:
932:
926:
923:
920:
916:
912:
909:
906:
900:
894:
887:
886:
885:
882:
868:
848:
826:
822:
801:
781:
761:
757:
750:
747:
744:
718:
712:
709:
689:
669:
649:
626:
620:
617:
597:
577:
554:
551:
548:
545:
542:
539:
536:
533:
530:
507:
487:
459:
455:
451:
448:
440:
437:
434:
430:
426:
420:
414:
407:
406:
405:
391:
383:
367:
360:is odd, and
347:
327:
307:
284:
278:
275:
255:
244:Constructions
243:
241:
224:
218:
215:
207:
189:
186:
183:
157:
151:
148:
140:
122:
119:
116:
90:
84:
81:
73:
55:
52:
49:
34:
32:
30:
27:is a type of
26:
19:
1698:
1694:
1683:
1625:
1526:
1476:
1474:
1464:
1414:
953:
883:
774:. Replacing
479:
340:are primes,
247:
72:Hamming code
38:
24:
22:
1529:(named for
520:in the set
29:cyclic code
1745:Categories
1677:References
1660:⌋
1643:−
1634:⌊
1490:≡
1366:⟩
1360:−
1347:⟨
1191:−
1163:∗
1123:∗
965:∈
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