33:
258:
1418:
The classical way to recover from failures in storage systems was to use replication. However, replication incurs significant overhead in terms of wasted bytes. Therefore, increasingly large storage systems, such as those used in data centers use erasure-coded storage. The most common form of erasure
1058:
Note that Alice cannot encode her telephone number in just one err-mail, because it contains six characters, and that the maximum length of one err-mail message is five characters. If she sent her phone number in pieces, asking Bob to acknowledge receipt of each piece, at least four messages would
236:
While technically RAID can be seen as a kind of erasure code, "RAID" is generally applied to an array attached to a single host computer (which is a single point of failure), while "erasure coding" generally implies multiple hosts, sometimes called a
1387:
symbol message into a practically infinite encoded form, i.e., they can generate an arbitrary amount of redundancy symbols that can all be used for error correction. Receivers can start decoding after they have received slightly more than
1398:
address the issue of rebuilding (also called repairing) lost encoded fragments from existing encoded fragments. This issue occurs in distributed storage systems where communication to maintain encoded redundancy is a problem.
1481:, 10 MB of user data is divided into ten 1MB blocks. Then, four additional 1 MB parity blocks are created to provide redundancy. This can tolerate up to 4 concurrent failures. The storage overhead here is 14/10 = 1.4X.
1491:
Initially, erasure codes were used to reduce the cost of storing "cold" (rarely-accessed) data efficiently; but erasure codes can also be used to improve performance serving "hot" (more-frequently-accessed) data.
2132:
is a near optimal extension to Luigi Rizzo's work that uses band matrices. Many parameters can be set, like the size of the width of the band and size of the finite field. It also successfully exploits the large
1484:
In the case of a fully replicated system, the 10 MB of user data will have to be replicated 4 times to tolerate up to 4 concurrent failures. The storage overhead in that case will be 50/10 = 5 times.
1495:
RAID N+M divides data blocks across N+M drives, and can recover all the data when any M drives fail. In particular, RAID 7.3 refers to triple-parity RAID, and can recover all the data when any 3 drives fail.
1163: − 1 'on the fly', i.e., distribute the workload evenly between transmission of the symbols. If the receiver wants to do his calculations 'on the fly', he can construct a new polynomial
655:
475:
540:
684: = 2, redundancy symbols may be created by sampling different points along the line between the two original symbols. This is pictured with a simple example, called err-mail:
395:
1915:
1019:
789:
845:
2034:
Dimakis, Alexandros G.; Godfrey, P. Brighten; Wu, Yunnan; Wainwright, Martin J.; Ramchandran, Kannan (September 2010). "Network Coding for
Distributed Storage Systems".
915:
880:
276:
1790:
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1059:
have to be sent anyway (two from Alice, and two acknowledgments from Bob). So the erasure code in this example, which requires five messages, is quite economical.
1488:
This gives an idea of the lower storage overhead of erasure-coded storage compared to full replication and thus the attraction in today's storage systems.
1034:
923:
222:
As of 2023, modern data storage systems can be designed to tolerate the complete failure of a few disks without data loss, using one of 3 approaches:
1952:
1407:
Erasure coding is now standard practice for reliable data storage. In particular, various implementations of Reed-Solomon erasure coding are used by
1864:
238:
2140:
2005:
1062:
This example is a little bit contrived. For truly generic erasure codes that work over any data set, we would need something other than the
315:
code word symbols are sufficient to recover the original message (i.e., they have optimal reception efficiency). Optimal erasure codes are
1780:. The third Sino-French Workshop on Information and Communication Technologies, SIFWICT 2015, Jun 2015, Nantes, France. ffhal-01162047f
1919:
1055:(5)) he has received. Bob can perform this procedure using any two err-mails, so the erasure code in this example has a rate of 40%.
1423:, an advanced mathematics formula used to enable regeneration of missing data from pieces of known data, called parity blocks. In a (
1420:
1369:
trade correction capabilities for computational complexity: practical algorithms can encode and decode with linear time complexity.
294:
116:
244:
Compared to block-level RAID systems, object storage erasure coding has some significant differences that make it more resilient.
1839:
1144: − 1). The receiver can now also use polynomial interpolation to recover the lost packets, provided he receives
1615:
316:
1518:
54:
575:
2149:
Developed in 1996, was the first use of FEC "Forward Error correction" on the
Internet. It was first used commercially to *
1671:
97:
212:
50:
69:
2089:
2123:
407:
76:
1567:
1753:
1555:
487:
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is a Free
Software library implementing Reed-Solomon and Cauchy erasure code techniques with SIMD optimisations.
1995:
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141:
83:
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43:
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65:
1943:. 2016. p. 2: section "Abstract". p. 9: section "Systematic codes". p. 12: section "Regenerating codes".
1868:
1609:
1550:
144:(FEC) code under the assumption of bit erasures (rather than bit errors), which transforms a message of
1940:
1725:
1395:
956:
726:
794:
2058:
1102:
208:
1636:(differs in that the original secret is encrypted and obscured until the decode quorum is reached)
1152:
is less than 2, where b is the number of bits in a symbol, then multiple polynomials can be used.
2134:
2071:
2043:
1328:
708:
Instead of asking Bob to acknowledge the messages she sends, Alice devises the following scheme.
1033:
922:
1365:
symbols to recover the message (where ε>0). Reducing ε can be done at the cost of CPU time.
2001:
1696:
1639:
885:
850:
1591:
2063:
545:
2137:
size of modern CPUs. How it compares to the near optimal codes mentioned above is unknown.
1335:
241:(RAIS). The erasure code allows operations to continue when any one of those hosts stops.
201:
90:
1599:, an MDS code outperforming Reed–Solomon in the maximal number of redundant packets, see
1029:
are the two parts of the telephone number. Now suppose Bob receives "D=777" and "E=851".
1633:
1347:
1283:
and evaluate it to find the lost packets. So both the sender and the receiver require
2160:
2075:
2021:
1530:
1513:
1408:
1372:
691:
687:
207:
There are many different erasure coding schemes. The most popular erasure codes are
133:
1545:
1535:
1324:
1083:
1415:
built into Linux, Microsoft Azure, Facebook cold storage, and
Backblaze Vaults.
216:
197:
32:
1981:
941:(5), and then transmits three redundant messages: "C=703", "D=777" and "E=851".
1814:
1582:
328:
2093:
2067:
152:
symbols such that the original message can be recovered from a subset of the
17:
1777:
569:, is erased, it can be easily recovered by summing the remaining variables:
169:
2150:
1097:, but usually a power of 2. The sender numbers the data symbols from 0 to
2151:
stream live video of Sir Arthur C. Clarke in Sri Lanka to UIUC in
Indiana
1577:
1540:
1791:"Understanding IBM Spectrum Scale Erasure Code Edition fault tolerance"
1776:
Dimitri Pertin, Alexandre van Kempen, Benoît
Parrein, Nicolas Normand.
1459:) chunks are available, one can recover the entire data. This means a (
542:
is now consistent with regard to the checksum. If one of these values,
1965:
1889:
1840:"Erasure coding vs Raid as a data protection method | Computer Weekly"
1412:
660:
1994:
Xia, Mingyuan; Saxena, Mohit; Blaum, Mario; Pease, David A. (2015).
1586:
1039:
Bob can reconstruct Alice's phone number by computing the values of
2048:
1672:"RAID vs. Erasure Coding. What's the Difference? | Blog | Xinnor"
2129:
1982:"Backblaze Open-sources Reed-Solomon Erasure Coding Source Code"
1571:
1504:
Here are some examples of implementations of the various codes:
1478:
1279:) can be calculated. So we have enough data points to construct
663:
is a widely-used application of the parity check erasure code.
184:. The recovery algorithm expects that it is known which of the
180:
denotes the number of symbols required for recovery, is called
251:
26:
1477:
In RS (10, 4) code, which is used in
Facebook for their
2090:"home [Erasure Coding for Distributed Storage Wiki]"
720:= 629, and sends 2 messages – "A=555" and "B=629" – to Bob.
1941:"Erasure Coding for Big-data Systems: Theory and Practice"
1101: − 1 and sends them. He then constructs a
2126:
by Luigi Rizzo describes optimal erasure correction codes
694:
using err-mail. Err-mail works just like e-mail, except
2117:
272:
1815:"Object Storage Erasure Coding vs. Block Storage RAID"
1342:
symbols can be found copied, unencoded, as one of the
650:{\displaystyle v_{e}=-\sum _{i=1,i\neq e}^{k+1}v_{i}.}
1916:"Why erasure coding is the future of data resiliency"
959:
888:
853:
797:
729:
578:
548:
490:
410:
351:
2143:
for regenerating codes and rebuilding erasure codes.
1676:
The
Fastest and Most Reliable Software RAID | Xinnor
1247:
has been received successfully. Secondly, if symbol
1078:
The linear construction above can be generalized to
2146:
1447:. The coding is done such that as long as at least
267:
may be too technical for most readers to understand
57:. Unsourced material may be challenged and removed.
1865:"Erasure Code: RAID As It Should Be – Huawei BLOG"
1013:
909:
874:
839:
783:
712:She breaks her telephone number up into two parts
649:
561:
534:
469:
389:
1403:Applications of erasure coding in storage systems
307:Optimal erasure codes have the property that any
1435:data blocks, called "chunks", are encoded into (
1082:. Additionally, points are now computed over a
690:wants to send her telephone number (555629) to
148:symbols into a longer message (code word) with
1525:Near optimal fountain (rateless erasure) codes
1443:) chunks. The total set of chunks comprises a
701:Messages longer than 5 characters are illegal.
470:{\displaystyle v_{k+1}=-\sum _{i=1}^{k}v_{i}.}
1346:message symbols. (Erasure codes that support
397:, a checksum is computed and appended to the
8:
505:
491:
366:
352:
704:It is very expensive (similar to air-mail).
535:{\displaystyle \{v_{i}\}_{1\leq i\leq k+1}}
1953:"Erasure Encoding—Practice and Principles"
1935:
1933:
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1976:
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390:{\displaystyle \{v_{i}\}_{1\leq i\leq k}}
369:
359:
350:
295:Learn how and when to remove this message
279:, without removing the technical details.
129:Code added to allow recovery of lost data
117:Learn how and when to remove this message
1914:Bhaskaran, Dinesh Kumar (July 6, 2018).
1666:
1664:
1662:
1660:
1467:) RS-encoded storage can tolerate up to
2124:Software FEC in computer communications
2036:IEEE Transactions on Information Theory
1726:"RAID vs Erasure Coding vs Replication"
1656:
333:Parity check is the special case where
1148:symbols successfully. If the order of
239:Redundant Array of Inexpensive Servers
2147:ECIP "Erasure Code Internet Protocol"
1997:A Tale of Two Erasure Codes in {HDFS}
1747:
1745:
1719:
1717:
1323:, with code words constructed over a
1255:has been received successfully, then
698:About half of all the mail gets lost.
277:make it understandable to non-experts
7:
1778:"Comparison of RAID-6 Erasure Codes"
1311:) space for operating 'on the fly'.
55:adding citations to reliable sources
2141:Coding for Distributed Storage wiki
1697:"Ceph.io — Erasure Coding in Ceph"
1419:coding used in storage systems is
723:She constructs a linear function,
25:
1597:Erasure Resilient Systematic Code
1334:Most practical erasure codes are
1155:The sender can construct symbols
1014:{\displaystyle f(i)=a+(b-a)(i-1)}
784:{\displaystyle f(i)=a+(b-a)(i-1)}
2022:""Triple-parity RAID and beyond"
1032:
921:
840:{\displaystyle f(i)=555+74(i-1)}
317:maximum distance separable codes
256:
31:
1616:maximum distance separable code
1319:This process is implemented by
1089:First we choose a finite field
196:Erasure coding was invented by
42:needs additional citations for
1519:Low-density parity-check codes
1350:never use a systematic code).
1191:was received successfully and
1008:
996:
993:
981:
969:
963:
898:
892:
863:
857:
834:
822:
807:
801:
778:
766:
763:
751:
739:
733:
341: + 1. From a set of
1:
213:Low-density parity-check code
2092:. 2017-07-31. Archived from
1338:-- each one of the original
1199:) = 0 when symbol
1863:Kruth, Peter (2023-10-04).
1724:Lee, Brandon (2023-12-26).
945:Bob knows that the form of
2183:
1508:Near optimal erasure codes
1431:) RS code, a given set of
1381:near-optimal erasure codes
1379:) are notable examples of
1367:Near-optimal erasure codes
1359:Near-optimal erasure codes
1354:Near-optimal erasure codes
1239:) = 0 if symbol
1207:was not received. Now let
1124:) is equal to data symbol
326:
1754:"RAID Vs. Erasure Coding"
1361:require (1 + ε)
1315:Real world implementation
680:In the simple case where
2068:10.1109/TIT.2010.2054295
1758:www.networkcomputing.com
1628:Forward error correction
1231:). Firstly we know that
1080:polynomial interpolation
929:She computes the values
910:{\displaystyle f(2)=629}
875:{\displaystyle f(1)=555}
142:forward error correction
1383:. They can transform a
1299:)) operations and only
1093:with order of at least
667:Polynomial oversampling
1645:Binary erasure channel
1421:Reed-Solomon (RS) code
1377:rateless erasure codes
1015:
911:
876:
841:
785:
651:
633:
563:
536:
484: + 1 values
471:
453:
391:
156:symbols. The fraction
1562:Optimal erasure codes
1103:(Lagrange) polynomial
1016:
912:
877:
842:
786:
652:
595:
564:
562:{\displaystyle v_{e}}
537:
472:
433:
392:
248:Optimal erasure codes
2000:. pp. 213–226.
1966:"Erasure Coding 101"
1890:"Erasure Coding 101"
1271:) −
1223:) −
957:
886:
851:
795:
727:
576:
546:
488:
408:
349:
182:reception efficiency
51:improve this article
1605:RS(9,2) with 3 bits
1601:RS(4,2) with 2 bits
1307: −
1295: −
672:Example: Err-mail (
209:Reed-Solomon coding
1922:on August 7, 2020.
1844:ComputerWeekly.com
1610:Regenerating Codes
1592:Reed–Solomon codes
1396:Regenerating codes
1329:Vandermonde matrix
1321:Reed–Solomon codes
1011:
907:
872:
837:
781:
647:
559:
532:
467:
387:
215:(LDPC codes), and
188:symbols are lost.
2007:978-1-931971-20-1
1640:Spelling alphabet
1392:encoded symbols.
1047:from the values (
305:
304:
297:
127:
126:
119:
101:
16:(Redirected from
2174:
2105:
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2102:
2101:
2086:
2080:
2079:
2061:
2051:
2042:(9): 4539–4551.
2031:
2025:
2020:Adam Leventhal.
2018:
2012:
2011:
1991:
1985:
1978:
1969:
1962:
1956:
1950:
1944:
1939:Rashmi Vinayak.
1937:
1924:
1923:
1918:. Archived from
1911:
1905:
1904:
1902:
1901:
1886:
1880:
1879:
1877:
1876:
1867:. Archived from
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1854:
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1805:
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1712:
1711:
1709:
1708:
1693:
1687:
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1684:
1683:
1668:
1574:storage systems.
1556:Triangular codes
1336:systematic codes
1203: <
1128:. He then sends
1036:
1020:
1018:
1017:
1012:
925:
916:
914:
913:
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790:
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59:
35:
27:
21:
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2177:
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2173:
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2171:
2157:
2156:
2114:
2109:
2108:
2099:
2097:
2088:
2087:
2083:
2059:10.1.1.117.6892
2033:
2032:
2028:
2019:
2015:
2008:
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1752:O'Reilly, Jim.
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1405:
1375:(also known as
1356:
1317:
1076:
955:
954:
884:
883:
849:
848:
793:
792:
791:, in this case
725:
724:
678:
676: = 2)
669:
634:
579:
574:
573:
549:
544:
543:
504:
494:
486:
485:
454:
411:
406:
405:
401:source values:
365:
355:
347:
346:
331:
325:
301:
290:
284:
281:
273:help improve it
270:
261:
257:
250:
202:Gustave Solomon
194:
172:. The fraction
130:
123:
112:
106:
103:
60:
58:
48:
36:
23:
22:
15:
12:
11:
5:
2180:
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2127:
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2113:
2112:External links
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1373:Fountain codes
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1352:
1348:secret sharing
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1313:
1263:) =
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66:"Erasure code"
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2167:Coding theory
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1748:
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1718:
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1702:
1698:
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1588:
1584:
1581:
1579:
1576:
1573:
1569:
1566:
1565:
1561:
1557:
1554:
1552:
1551:Network Codes
1549:
1547:
1544:
1542:
1539:
1537:
1534:
1532:
1531:Fountain code
1529:
1528:
1524:
1520:
1517:
1515:
1514:Tornado codes
1512:
1511:
1507:
1505:
1499:
1497:
1493:
1489:
1486:
1482:
1480:
1476:
1472:
1470:
1466:
1462:
1458:
1455: +
1454:
1450:
1446:
1442:
1439: +
1438:
1434:
1430:
1426:
1422:
1416:
1414:
1410:
1409:Apache Hadoop
1402:
1400:
1397:
1393:
1391:
1386:
1382:
1378:
1374:
1370:
1368:
1364:
1360:
1353:
1351:
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1345:
1341:
1337:
1332:
1330:
1326:
1322:
1314:
1312:
1310:
1306:
1302:
1298:
1294:
1290:
1286:
1282:
1278:
1274:
1270:
1266:
1262:
1258:
1254:
1251: ≥
1250:
1246:
1242:
1238:
1234:
1230:
1226:
1222:
1218:
1214:
1210:
1206:
1202:
1198:
1194:
1190:
1186:
1182:
1178:
1174:
1170:
1166:
1162:
1158:
1153:
1151:
1147:
1143:
1139:
1135:
1131:
1127:
1123:
1119:
1115:
1111:
1107:
1104:
1100:
1096:
1092:
1087:
1085:
1081:
1073:
1071:
1069:
1065:
1060:
1056:
1054:
1050:
1046:
1042:
1037:
1035:
1030:
1028:
1024:
1005:
1002:
999:
990:
987:
984:
978:
975:
972:
966:
960:
952:
948:
940:
936:
932:
928:
927:
926:
924:
904:
901:
895:
889:
869:
866:
860:
854:
831:
828:
825:
819:
816:
813:
810:
804:
798:
775:
772:
769:
760:
757:
754:
748:
745:
742:
736:
730:
722:
719:
715:
711:
710:
709:
703:
700:
697:
696:
695:
693:
689:
685:
683:
675:
671:
666:
664:
662:
644:
639:
635:
629:
626:
623:
618:
615:
612:
609:
606:
603:
600:
596:
592:
589:
584:
580:
572:
571:
570:
554:
550:
527:
524:
521:
518:
515:
512:
509:
499:
495:
483:
464:
459:
455:
449:
444:
441:
438:
434:
430:
427:
422:
419:
416:
412:
404:
403:
402:
400:
382:
379:
376:
373:
370:
360:
356:
344:
340:
336:
330:
322:
320:
319:(MDS codes).
318:
314:
310:
299:
296:
288:
278:
274:
268:
265:This section
263:
254:
253:
247:
245:
242:
240:
231:
228:
225:
224:
223:
220:
218:
214:
210:
205:
203:
199:
191:
189:
187:
183:
179:
175:
171:
167:
163:
160: =
159:
155:
151:
147:
143:
139:
135:
134:coding theory
121:
118:
110:
99:
96:
92:
89:
85:
82:
78:
75:
71:
68: –
67:
63:
62:Find sources:
56:
52:
46:
45:
40:This article
38:
34:
29:
28:
19:
18:Erasure codes
2098:. Retrieved
2094:the original
2084:
2039:
2035:
2029:
2016:
1996:
1989:
1960:
1948:
1920:the original
1909:
1898:. Retrieved
1896:. 2022-04-25
1893:
1884:
1873:. Retrieved
1869:the original
1858:
1847:. Retrieved
1843:
1834:
1823:. Retrieved
1821:. 2021-07-27
1818:
1809:
1798:. Retrieved
1794:
1785:
1772:
1761:. Retrieved
1757:
1733:. Retrieved
1729:
1705:. Retrieved
1703:. 2014-04-07
1700:
1691:
1680:. Retrieved
1678:. 2023-09-03
1675:
1546:Raptor codes
1536:Online codes
1503:
1494:
1490:
1487:
1483:
1474:
1473:
1468:
1464:
1460:
1456:
1452:
1448:
1444:
1440:
1436:
1432:
1428:
1424:
1417:
1406:
1394:
1389:
1384:
1380:
1376:
1371:
1366:
1362:
1358:
1357:
1343:
1339:
1333:
1325:finite field
1318:
1308:
1304:
1300:
1296:
1292:
1288:
1284:
1280:
1276:
1272:
1268:
1264:
1260:
1256:
1252:
1248:
1244:
1240:
1236:
1232:
1228:
1224:
1220:
1216:
1212:
1208:
1204:
1200:
1196:
1192:
1188:
1184:
1183:) if symbol
1180:
1176:
1172:
1168:
1167:, such that
1164:
1160:
1156:
1154:
1149:
1145:
1141:
1137:
1133:
1129:
1125:
1121:
1117:
1113:
1109:
1105:
1098:
1094:
1090:
1088:
1084:finite field
1077:
1074:General case
1067:
1063:
1061:
1057:
1052:
1048:
1044:
1040:
1038:
1031:
1026:
1022:
950:
946:
944:
938:
934:
930:
920:
847:, such that
717:
713:
707:
686:
681:
679:
673:
659:
481:
479:
398:
342:
338:
334:
332:
323:Parity check
312:
308:
306:
291:
282:
266:
243:
235:
221:
206:
195:
185:
177:
173:
165:
161:
157:
153:
149:
145:
138:erasure code
137:
131:
113:
104:
94:
87:
80:
73:
61:
49:Please help
44:verification
41:
1795:www.ibm.com
1471:failures.
1112:) of order
480:The set of
311:out of the
285:August 2023
226:Replication
217:Turbo codes
198:Irving Reed
107:August 2023
2100:2023-08-20
2049:cs/0702015
1900:2024-09-18
1894:MinIO Blog
1875:2024-09-18
1849:2024-09-18
1825:2024-09-18
1819:MinIO Blog
1800:2024-09-18
1763:2024-09-18
1735:2024-09-18
1707:2024-09-18
1682:2024-09-18
1651:References
1614:any other
1583:Tahoe-LAFS
1570:: used in
1116:such that
329:Parity bit
327:See also:
77:newspapers
2076:260559901
2054:CiteSeerX
1585:includes
1070:) given.
1003:−
988:−
937:(4), and
829:−
773:−
758:−
616:≠
597:∑
593:−
519:≤
513:≤
435:∑
431:−
380:≤
374:≤
204:in 1960.
170:code rate
2161:Category
2135:register
2118:Jerasure
1730:BDRSuite
1622:See also
1578:Parchive
1541:LT codes
1500:Examples
1475:Example:
1451:out of (
1327:using a
1136:), ...,
1051:(4) and
1021:, where
176:, where
2024:. 2009.
1984:. 2015.
1968:. 2022.
1955:. 2016.
1701:ceph.io
1463:,
1427:,
716:= 555,
345:values
337:=
335:n
271:Please
192:History
91:scholar
2130:Feclib
2074:
2056:
2004:
1630:codes.
1568:Parity
1445:stripe
1413:RAID-6
1411:, the
661:RAID 5
93:
86:
79:
72:
64:
2072:S2CID
2044:arXiv
1243:<
1187:<
953:) is
933:(3),
688:Alice
140:is a
136:, an
98:JSTOR
84:books
2002:ISBN
1587:zfec
1572:RAID
1479:HDFS
1215:) =
1175:) =
1043:and
1025:and
882:and
229:RAID
200:and
174:k’/k
70:news
2064:doi
1603:or
1159:to
905:629
870:555
814:555
692:Bob
275:to
132:In
53:by
2163::
2070:.
2062:.
2052:.
2040:56
2038:.
1973:^
1928:^
1892:.
1842:.
1817:.
1793:.
1756:.
1744:^
1728:.
1716:^
1699:.
1674:.
1659:^
1331:.
1086:.
820:74
219:.
211:,
178:k’
2153:.
2103:.
2078:.
2066::
2046::
2010:.
1903:.
1878:.
1852:.
1828:.
1803:.
1766:.
1738:.
1710:.
1685:.
1469:m
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1390:k
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1303:(
1301:O
1297:k
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1287:(
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1006:1
1000:i
997:(
994:)
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979:+
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970:)
967:i
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902:=
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896:2
893:(
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864:)
861:1
858:(
855:f
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832:1
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823:(
817:+
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802:(
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779:)
776:1
770:i
767:(
764:)
761:a
755:b
752:(
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714:a
682:k
674:k
645:.
640:i
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607:1
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510:1
506:}
500:i
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492:{
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186:n
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114:(
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95:·
88:·
81:·
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47:.
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