New Constructions of Binary LRCs With Disjoint Repair Groups and Locality 3 Using Existing LRCs

Kim, Chanki; No, Jong-Seon

doi:10.1109/lcomm.2019.2892950

Cited by 12 publications

(7 citation statements)

References 10 publications

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“…. , 16)) as given by Definition 9, where also q = 16, C 2 ⊂ C 1 ⊂ C 0 are extended RS codes over GF (16) such that C 0 is a [16,15,2] code, C 1 is a [16,13,4] code, C 2 is a [16,11,6] code, V 0 is a [256, 243,14] code over (GF(16)) 11 , V 1 is a [256, 236,21] code over GF(256) and V 2 is a [256, 216,41] code over GF(256). By Corollary 16, C (II) has dimension 3577 and minimum distance 82, i.e., it is a [4096, 3577, 82] code over GF(16) whose minimum distance is significantly larger than the one of C (I) and its dimension is also larger.…”

Section: Proof: By Theorem 15 and Sincementioning

confidence: 99%

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: 99%

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

Blaum

2020

IEEE Trans. Inform. Theory

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“…If we regard m as the number of rows of H and let u = m−l, then H G can be represented as an u×n matrix. The function of H G is that determines the minimum distance of C. Moreover, the parity-check matrix H can be represented as follows [10]- [12]:…”

Section: Construction Of Lrcs With Disjoint Repair Groupsmentioning

confidence: 99%

“…Yang et al studied the locality of low dimensional ternary optimal codes in [7]- [9]. References [10]- [12] utilized t-spreads to construct binary LRCs of 6 with disjoint repair groups. Hao et al [13] proposed a class of LRCs with q ≥ r − 1, q = 2 n−k and d = 4 and found four classes of binary LRCs, which reach the Singleton-like bound.…”

Section: Introductionmentioning

confidence: 99%

Constructions and Some Search Results of Ternary LRCs with <i>d</i> = 6

Zheng

et al. 2021

IEICE Trans. Fundamentals

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Locally repairable codes (LRCs) are a type of new erasure codes designed for modern distributed storage systems (DSSs). In order to obtain ternary LRCs of distance 6, firstly, we propose constructions with disjoint repair groups and construct several families of LRCs with 1 ≤ r ≤ 6, where codes with 3 ≤ r ≤ 6 are obtained through a search algorithm. Then, we propose a new method to extend the length of codes without changing the distance. By employing the methods such as expansion and deletion, we obtain more LRCs from a known LRC. The resulting LRCs are optimal or near optimal in terms of the Cadambe-Mazumdar (C-M) bound.

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“…In [16], a construction for LRCs was presented via puncturing anti-codes from generator matrices of Simplex codes, some optimal LRCs with small localities and low rates were derived. Using block-puncturing methods on generator matrices of Simplex codes, authors of [24] investigated minimum distance, locality, availability, joint information locality, and joint information availability of related LRCs.…”

Section: B Lrcs Constructed By Dual Puncturing Known Codesmentioning

confidence: 99%

New Constructions of Short Length Binary Locally Repairable Codes

Yang

2020

IEEE Access

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In this short paper, our main objective is to construct binary locally repairable codes (LRCs) with good properties. Two constructions of LRCs with short lengthes are proposed. The first one is that define a u-linearly independent set (u-LIS) of an LRC with disjoint repair groups (DRGs) and enlarge it into another one with bigger size to construct new LRCs. The second is puncturing check matrices of known codes to construct new LRCs. As an application, many new binary LRCs are constructed from distance optimal linear codes, which are also locality optimal according to the Cadambe-Mazumdar (C-M) bound. INDEX TERMS Cadambe-Mazumdar bound, distance optimal linear code, locally repairable code.

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New Constructions of Binary LRCs With Disjoint Repair Groups and Locality 3 Using Existing LRCs

Cited by 12 publications

References 10 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

Constructions and Some Search Results of Ternary LRCs with <i>d</i> = 6

New Constructions of Short Length Binary Locally Repairable Codes

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