Lan Ma;Liyang Zhou;Shaoteng Liu;Xiangyu Chen;Qifu Sun
{"title":"带有两个奇偶校验的新系统 MDS 阵列代码","authors":"Lan Ma;Liyang Zhou;Shaoteng Liu;Xiangyu Chen;Qifu Sun","doi":"10.1109/OJCOMS.2024.3468873","DOIUrl":null,"url":null,"abstract":"Row-diagonal parity (RDP) code is a classical \n<inline-formula> <tex-math>$(k+2,~k)$ </tex-math></inline-formula>\n systematic maximum distance separable (MDS) array code with \n<inline-formula> <tex-math>$k \\leq L-1$ </tex-math></inline-formula>\n under sub-packetization level \n<inline-formula> <tex-math>$l = L-1$ </tex-math></inline-formula>\n, where L is a prime integer. When \n<inline-formula> <tex-math>$k = L-1$ </tex-math></inline-formula>\n, its encoding requires \n<inline-formula> <tex-math>$2-{}\\frac {2}{k}$ </tex-math></inline-formula>\n XORs per original data bit, which exactly achieves theoretical optimal lower bound. In this paper, we present three new constructions of \n<inline-formula> <tex-math>$(k+2,~k)$ </tex-math></inline-formula>\n systematic MDS array codes. First, under sub-packetization level \n<inline-formula> <tex-math>$l = 4$ </tex-math></inline-formula>\n, we novelly design a \n<inline-formula> <tex-math>$(17,~15)$ </tex-math></inline-formula>\n array code \n<inline-formula> <tex-math>${\\mathcal {C}}_{1}$ </tex-math></inline-formula>\n, where k can reach the largest possible value to satisfy the MDS property. Moreover, when \n<inline-formula> <tex-math>$k \\leq 7$ </tex-math></inline-formula>\n, the encoding complexity of its subcodes can exactly achieve the theoretical optimal \n<inline-formula> <tex-math>$2-{}\\frac {2}{k}$ </tex-math></inline-formula>\n XORs per original data bit, and likewise, the decoding complexity of the subcodes with \n<inline-formula> <tex-math>$k \\leq 4$ </tex-math></inline-formula>\n is also exactly optimal. Under sub-packetization level \n<inline-formula> <tex-math>$l = L-1$ </tex-math></inline-formula>\n with certain primes L, the second construction yields an MDS array code \n<inline-formula> <tex-math>${\\mathcal {C}}_{2}$ </tex-math></inline-formula>\n with \n<inline-formula> <tex-math>$k \\leq {}\\frac {L(L-1)}{2}$ </tex-math></inline-formula>\n, and the encoding complexity of \n<inline-formula> <tex-math>${\\mathcal {C}}_{2}$ </tex-math></inline-formula>\n is also exactly optimal for \n<inline-formula> <tex-math>$k = L-1$ </tex-math></inline-formula>\n, \n<inline-formula> <tex-math>$2L-3$ </tex-math></inline-formula>\n. Furthermore, based on bit permutation, the third MDS array code \n<inline-formula> <tex-math>${\\mathcal {C}}_{3}$ </tex-math></inline-formula>\n is obtained with \n<inline-formula> <tex-math>$k \\leq L(L-1)$ </tex-math></inline-formula>\n under sub-packetization level \n<inline-formula> <tex-math>$l = 2(L-1)$ </tex-math></inline-formula>\n with certain primes L. In particular, as an extension of \n<inline-formula> <tex-math>${\\mathcal {C}}_{2}$ </tex-math></inline-formula>\n, \n<inline-formula> <tex-math>${\\mathcal {C}}_{3}$ </tex-math></inline-formula>\n exactly achieves the optimal encoding complexity for \n<inline-formula> <tex-math>$k = 2(2L-3)$ </tex-math></inline-formula>\n, which does not hold for other array codes in the literature.","PeriodicalId":33803,"journal":{"name":"IEEE Open Journal of the Communications Society","volume":null,"pages":null},"PeriodicalIF":6.3000,"publicationDate":"2024-09-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=10695780","citationCount":"0","resultStr":"{\"title\":\"New Systematic MDS Array Codes With Two Parities\",\"authors\":\"Lan Ma;Liyang Zhou;Shaoteng Liu;Xiangyu Chen;Qifu Sun\",\"doi\":\"10.1109/OJCOMS.2024.3468873\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"Row-diagonal parity (RDP) code is a classical \\n<inline-formula> <tex-math>$(k+2,~k)$ </tex-math></inline-formula>\\n systematic maximum distance separable (MDS) array code with \\n<inline-formula> <tex-math>$k \\\\leq L-1$ </tex-math></inline-formula>\\n under sub-packetization level \\n<inline-formula> <tex-math>$l = L-1$ </tex-math></inline-formula>\\n, where L is a prime integer. When \\n<inline-formula> <tex-math>$k = L-1$ </tex-math></inline-formula>\\n, its encoding requires \\n<inline-formula> <tex-math>$2-{}\\\\frac {2}{k}$ </tex-math></inline-formula>\\n XORs per original data bit, which exactly achieves theoretical optimal lower bound. In this paper, we present three new constructions of \\n<inline-formula> <tex-math>$(k+2,~k)$ </tex-math></inline-formula>\\n systematic MDS array codes. First, under sub-packetization level \\n<inline-formula> <tex-math>$l = 4$ </tex-math></inline-formula>\\n, we novelly design a \\n<inline-formula> <tex-math>$(17,~15)$ </tex-math></inline-formula>\\n array code \\n<inline-formula> <tex-math>${\\\\mathcal {C}}_{1}$ </tex-math></inline-formula>\\n, where k can reach the largest possible value to satisfy the MDS property. Moreover, when \\n<inline-formula> <tex-math>$k \\\\leq 7$ </tex-math></inline-formula>\\n, the encoding complexity of its subcodes can exactly achieve the theoretical optimal \\n<inline-formula> <tex-math>$2-{}\\\\frac {2}{k}$ </tex-math></inline-formula>\\n XORs per original data bit, and likewise, the decoding complexity of the subcodes with \\n<inline-formula> <tex-math>$k \\\\leq 4$ </tex-math></inline-formula>\\n is also exactly optimal. Under sub-packetization level \\n<inline-formula> <tex-math>$l = L-1$ </tex-math></inline-formula>\\n with certain primes L, the second construction yields an MDS array code \\n<inline-formula> <tex-math>${\\\\mathcal {C}}_{2}$ </tex-math></inline-formula>\\n with \\n<inline-formula> <tex-math>$k \\\\leq {}\\\\frac {L(L-1)}{2}$ </tex-math></inline-formula>\\n, and the encoding complexity of \\n<inline-formula> <tex-math>${\\\\mathcal {C}}_{2}$ </tex-math></inline-formula>\\n is also exactly optimal for \\n<inline-formula> <tex-math>$k = L-1$ </tex-math></inline-formula>\\n, \\n<inline-formula> <tex-math>$2L-3$ </tex-math></inline-formula>\\n. Furthermore, based on bit permutation, the third MDS array code \\n<inline-formula> <tex-math>${\\\\mathcal {C}}_{3}$ </tex-math></inline-formula>\\n is obtained with \\n<inline-formula> <tex-math>$k \\\\leq L(L-1)$ </tex-math></inline-formula>\\n under sub-packetization level \\n<inline-formula> <tex-math>$l = 2(L-1)$ </tex-math></inline-formula>\\n with certain primes L. In particular, as an extension of \\n<inline-formula> <tex-math>${\\\\mathcal {C}}_{2}$ </tex-math></inline-formula>\\n, \\n<inline-formula> <tex-math>${\\\\mathcal {C}}_{3}$ </tex-math></inline-formula>\\n exactly achieves the optimal encoding complexity for \\n<inline-formula> <tex-math>$k = 2(2L-3)$ </tex-math></inline-formula>\\n, which does not hold for other array codes in the literature.\",\"PeriodicalId\":33803,\"journal\":{\"name\":\"IEEE Open Journal of the Communications Society\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":6.3000,\"publicationDate\":\"2024-09-26\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"https://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=10695780\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"IEEE Open Journal of the Communications Society\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://ieeexplore.ieee.org/document/10695780/\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q1\",\"JCRName\":\"ENGINEERING, ELECTRICAL & ELECTRONIC\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"IEEE Open Journal of the Communications Society","FirstCategoryId":"1085","ListUrlMain":"https://ieeexplore.ieee.org/document/10695780/","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"ENGINEERING, ELECTRICAL & ELECTRONIC","Score":null,"Total":0}
Row-diagonal parity (RDP) code is a classical
$(k+2,~k)$
systematic maximum distance separable (MDS) array code with
$k \leq L-1$
under sub-packetization level
$l = L-1$
, where L is a prime integer. When
$k = L-1$
, its encoding requires
$2-{}\frac {2}{k}$
XORs per original data bit, which exactly achieves theoretical optimal lower bound. In this paper, we present three new constructions of
$(k+2,~k)$
systematic MDS array codes. First, under sub-packetization level
$l = 4$
, we novelly design a
$(17,~15)$
array code
${\mathcal {C}}_{1}$
, where k can reach the largest possible value to satisfy the MDS property. Moreover, when
$k \leq 7$
, the encoding complexity of its subcodes can exactly achieve the theoretical optimal
$2-{}\frac {2}{k}$
XORs per original data bit, and likewise, the decoding complexity of the subcodes with
$k \leq 4$
is also exactly optimal. Under sub-packetization level
$l = L-1$
with certain primes L, the second construction yields an MDS array code
${\mathcal {C}}_{2}$
with
$k \leq {}\frac {L(L-1)}{2}$
, and the encoding complexity of
${\mathcal {C}}_{2}$
is also exactly optimal for
$k = L-1$
,
$2L-3$
. Furthermore, based on bit permutation, the third MDS array code
${\mathcal {C}}_{3}$
is obtained with
$k \leq L(L-1)$
under sub-packetization level
$l = 2(L-1)$
with certain primes L. In particular, as an extension of
${\mathcal {C}}_{2}$
,
${\mathcal {C}}_{3}$
exactly achieves the optimal encoding complexity for
$k = 2(2L-3)$
, which does not hold for other array codes in the literature.
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