Partial MDS (PMDS) and Sector-Disk (SD) codes that tolerate the erasure of two random sectors

Blaum, M.; Plank, James S.; Schwartz, Moshe; Yaakobi, Eitan

doi:10.1109/isit.2014.6875142

Cited by 19 publications

(39 citation statements)

References 7 publications

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“…As done in Example 7, we may consider the average number of erasures that both codes can correct. Doing a Montecarlo simulation, we find out that C ( 7)), C 0 , C 1 and C 2 are like in C (II) , C 3 ⊂ C 2 is a [16,9,8] (16) by Corollary 16, hence, its minimum distance is smaller than the one of C (I) and considerably smaller than the one of C (II) . However, the average number of erasures it can correct is 369, much larger than the average of 184 erasures that C (II) can correct.…”

Section: Proof: By Theorem 15 and Sincementioning

confidence: 93%

“…Assume that t > 1. By Lemma 12, C(n, u) is the direct sum of C(n, u (0) ) and C(n, u ′ ), where u ′ is given by (9), so its dimension is the sum of the dimensions of C(n, u ′ ) and of C(n, u (0) ).…”

Section: Definition and Properties Of T-level Eii Codesmentioning

confidence: 99%

“…, 2, 4, 8)) as given by Definition 9, where q = 16, C 2 ⊂ C 1 ⊂ C 0 are extended RS codes over GF (16) such that C 0 is a [16,14,3] code, C 1 is a [16,12,5] code, C 2 is a [16,8,9] code, V 1 is a [256, 255, 2] code over (GF(16)) 8 and V 2 is a [256, 254, 3] code over GF(256). By Corollary 16, C (I) has dimension 3576 and minimum distance 9, i.e., it is a [4096, 3576,9] code over GF (16). In particular, it is a special case of Example 4 in [36], where, following the notation in that example, µ = 3, d ′ 3 = 9, d ′ 2 = 5, d ′ 1 = 3, δ 3 = 2 and δ 2 = 3.…”

Section: Proof: By Theorem 15 and Sincementioning

confidence: 99%

“…Next, assume that t > 1 and by induction, assume that there is a systematic encoder for any (t − 1)-level EII code. Consider the (t − 1)-level EII code C(n, u ′ ), where u ′ is given by (9), and encode systematically the data D i,j for 1 v t − 1, 0 i (m −ŝ v+1 ) − 1 and n − u v j n − u v−1 − 1 into an array C ′ ∈ C(n, u ′ ). Denote by P ′ i,j the parity symbols obtained as a result of this systematic encoder.…”

Section: Systematic Encoding and Decoding Of Eii Codesmentioning

confidence: 99%

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Extended Integrated Interleaved Codes Over Any Field With Applications to Locally Recoverable Codes

Blaum

2020

IEEE Trans. Inform. Theory

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Integrated Interleaved (II) and Extended Integrated Interleaved (EII) codes are a versatile alternative for Locally Recoverable (LRC) codes, since they require fields of relatively small size. II and EII codes are generally defined over Reed-Solomon type of codes. A new comprehensive definition of EII codes is presented, allowing for EII codes over any field, and in particular, over the binary field GF(2). The traditional definition of II and EII codes is shown to be a special case of the new definition. Improvements over previous constructions of LRC codes, in particular, for binary codes, are given, as well as cases meeting an upper bound on the minimum distance. Properties of the codes are presented as well, in particular, an iterative decoding algorithm on rows and columns generalizing the iterative decoding algorithm of product codes. Two applications are also discussed: one is finding a systematic encoding of EII codes such that the parity symbols have a balanced distribution on rows, and the other is the problem of ordering the symbols of an EII code such that the maximum length of a correctable burst is achieved.Constructions of LRC codes involve different issues and tradeoffs, like the size of the field and optimality criteria. The same is true for EPC codes, of which, as we have seen above, LRC codes are a special case. In particular, one goal is to keep the size of the required finite field small, since operations over a small field have less complexity than ones over a larger field due to the smaller look-up tables involved. For example, Integrated Interleaved (II) codes [32], [68] over GF(q), where q > max{m, n}, were proposed in [7] as LRC codes. The construction in [45], [46] (stair codes) reduces field size when failures are correlated. Extended Integrated Interleaved (EII) codes [8] unify product codes and II codes.As is the case with LRC codes, construction of EPC codes involves optimality issues. For example, LRC codes optimizing the minimum distance were presented in [67]. Except for special cases, II codes are not optimal as LRC codes, but the codes in [67] require a field of size at least mn. The same happens with EII codes: except for special cases, they do not optimize the minimum distance [8].There are stronger criteria for optimization than the minimum distance in LRC codes. For example, PMDS codes [5], [9], [19], [23], [33], [35] satisfy the Maximally Recoverable (MR) property [23], [25]. The definition of the MR property is extended for EPC codes in [25], but it turns out that EPC codes with the MR property are difficult to obtain. For example, in [25] it was proven that an EPC code EP(n, 1; n, 1; 1) (i.e., one vertical and one horizontal parity per column and row and one extra parity) with the MR property requires a field whose size is superlinear on n.The EII codes presented in [8], of which II codes [7], [68], [73], [76], [77] are special cases, in general assume a field GF(q) such that q > max{m, n} and the individual codes are RS type of codes. An exception occurs in [73], where b...

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Section: Proof: By Theorem 15 and Sincementioning

confidence: 93%

Section: Definition and Properties Of T-level Eii Codesmentioning

confidence: 99%

Section: Proof: By Theorem 15 and Sincementioning

confidence: 99%

Section: Systematic Encoding and Decoding Of Eii Codesmentioning

confidence: 99%

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Extended Integrated Interleaved Codes Over Any Field With Applications to Locally Recoverable Codes

Blaum

2020

IEEE Trans. Inform. Theory

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“…In [3]- [5], Blaum et al gave some constructions of PMDS codes which can tolerate any number of disk failures and two sector erasures, i.e. (r, 2)-PMDS codes.…”

Section: Introductionmentioning

confidence: 99%

Sector-disk codes and partial MDS codes with up to three global parities

Chen

Shum

et al. 2015

2015 IEEE International Symposium on Information Theory (ISIT)

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A new construction for sector-disk codes and partial MDS codes up to three sector erasures are proposed. In contrast to existing codes, which are based on the design of parity check matrix, our new code is based on the design of generator matrix. This new approach allows us to construct a code that requires a small field size. In particular, for the case when there is only one sector erasure, our field size requirement is independent of the number of sectors and is close to optimal. For the case when there are three sector erasures, we provide a condition which allows the code be constructed by computer search.

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Random construction of partial MDS codes

Neri

Horlemann-Trautmann

2019

Des. Codes Cryptogr.

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This work deals with partial MDS (PMDS) codes, a special class of locally repairable codes, used for distributed storage system. We first show that a known construction of these codes, using Gabidulin codes, can be extended to use any maximum rank distance code. Then we define a standard form for the generator matrices of PMDS codes and use this form to give an algebraic description of PMDS generator matrices. This implies that over a sufficiently large finite field a randomly chosen generator matrix in PMDS standard form generates a PMDS code with high probability. This also provides sufficient conditions on the field size for the existence of PMDS codes.

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Partial MDS (PMDS) and Sector-Disk (SD) codes that tolerate the erasure of two random sectors

Cited by 19 publications

References 7 publications

Extended Integrated Interleaved Codes Over Any Field With Applications to Locally Recoverable Codes

Extended Integrated Interleaved Codes Over Any Field With Applications to Locally Recoverable Codes

Sector-disk codes and partial MDS codes with up to three global parities

Random construction of partial MDS codes

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