Locality of optimal binary codes

Fu, Qiang; Li, Ruihu; Guo, Luobin; Lv, Liangdong

doi:10.1016/j.ffa.2017.08.013

Cited by 16 publications

(8 citation statements)

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“…Next take (16) with 1 i t − 1, then m−1 w=0 h r,w c w ∈ C i forŝ i+1 r ŝ i − 1, if and only if, since the parity-check matrix of C i is H u i , n , 0 as given by (4) and since c w ∈ C 0 , n−1 j=0 h v,j c w,j = 0 for 0 v u 0 − 1, if and only if, by (18) in Lemma 41, where H s , w , v is given by (4) and H 8,8,0 by (10). Since the number of parities in the II code C (8, (1, 1, 1, 1, 4, 4, 4, 7)) is 23, this is the rank of H(8, (1, 1, 1, 1, 4, 4, 4, 7)).…”

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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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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 take l = 5 in the example above, we get a binary linear LRC with parameters [15, 4, 8; 2] 2 . Moreover, we can obtain binary linear LRCs with parameters [20,8,8; 3] 2 and [24, 11, 8; 3] 2 by using the method in Construction IV.18. All of these examples are optimal with respect to bound (5).…”

Section: Discussion and Concluding Remarksmentioning

confidence: 99%

“…Another construction of binary LRCs with minimum distance at least 6 is given in [16], showing that some examples are optimal with respect to bound (2). In [8], the authors investigate the locality of MacDonald codes and generalized MacDonald codes, and propose some constructions of binary LRCs.…”

Section: Introductionmentioning

confidence: 99%

Optimal Binary Linear Locally Repairable Codes with Disjoint Repair Groups

Ma¹,

Ge²

2019

SIAM J. Discrete Math.

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In recent years, several classes of codes are introduced to provide some fault-tolerance and guarantee system reliability in distributed storage systems, among which locally repairable codes (LRCs for short) play an important role. However, most known constructions are over large fields with sizes close to the code length, which lead to the systems computationally expensive. Due to this, binary LRCs are of interest in practice.In this paper, we focus on binary linear LRCs with disjoint repair groups. We first derive an explicit bound for the dimension k of such codes, which can be served as a generalization of the bounds given in [11], [36], [37]. We also give several new constructions of binary LRCs with minimum distance d = 6 based on weakly independent sets and partial spreads, which are optimal with respect to our newly obtained bound. In particular, for locality r ∈ {2, 3} and minimum distance d = 6, we obtain the desired optimal binary linear LRCs with disjoint repair groups for almost all parameters.which is also called as Singleton-like bound since it degenerates to classical Singleton bound when r = k. Later, in [7], [19], the bound (1) is generalized to vector codes and nonlinear codes. Although it certainly holds for all LRCs, it is not tight in many cases. The tightness of the bound (1) is studied in [28], [34].We say an LRC is d-optimal if it satisfies bound (1) with equality for given n, k and r. For the case (r + 1)|n, d-optimal LRCs are constructed explicitly in [32] and [25] by using Reed-Solomon codes and Gabidulin codes respectively. However, both constructions are built over a finite field whose size is an exponential function of the code length n. In [30], for the same case (r + 1)|n, the authors construct a d-optimal code over a finite field of size sightly greater than n by using "good" polynomials. This construction can be extended to the case (r + 1) ∤ n with the minimum distance d ≥ n − k − ⌈ k r ⌉ + 1 which

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“…This bound is not tight over small fields [3], [4], especially over the binary field [5]. A bound taking field size into consideration was presented in [6], which is called Cadambe…”

Section: Introductionmentioning

confidence: 99%

“…Binary LRCs receive much more attentions, since they are easily implemented for no multiplications are needed in encoding, decoding and repair, see [2], [3], [5] and [7]- [19].…”

Section: Introductionmentioning

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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Locality of optimal binary codes

Cited by 16 publications

References 8 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

Optimal Binary Linear Locally Repairable Codes with Disjoint Repair Groups

New Constructions of Short Length Binary Locally Repairable Codes

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