Constructions and Properties of Linear Locally Repairable Codes

Ernvall, Toni; Westerbäck, Thomas; Freij-Hollanti, Ragnar; Hollanti, Camilla

doi:10.1109/tit.2015.2512864

Cited by 39 publications

(76 citation statements)

References 25 publications

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“…Several constructions of LRCs achieving the bound (2) already exist, for example in [6], [14], [15], [19], [20]. Many of these constructions require an alphabet size to be exponential in the code length.…”

Section: Achievability Resultsmentioning

confidence: 99%

Alphabet-Dependent Bounds for Linear Locally Repairable Codes Based on Residual Codes

Grezet

Freij-Hollanti

Westerbäck

et al. 2019

IEEE Trans. Inform. Theory

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Locally repairable codes (LRCs) have gained significant interest for the design of large distributed storage systems as they allow a small number of erased nodes to be recovered by accessing only a few others. Several works have thus been carried out to understand the optimal rate-distance tradeoff, but only recently the size of the alphabet has been taken into account. In this paper, a novel definition of locality is proposed to keep track of the precise number of nodes required for a local repair when the repair sets do not yield MDS codes. Then, a new alphabet-dependent bound is derived, which applies both to the new definition and the initial definition of locality. The new bound is based on consecutive residual codes and intrinsically uses the Griesmer bound. A special case of the bound yields both the extension of the Cadambe-Mazumdar bound and the Singleton-type bound for codes with locality (r, δ), implying that the new bound is at least as good as these bounds. Furthermore, an upper bound on the asymptotic rate-distance tradeoff of LRCs is derived, and yields the tightest known upper bound for large relative minimum distances. Achievability results are also provided by deriving the locality of the family of Simplex codes together with a few examples of optimal codes.

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Section: Achievability Resultsmentioning

confidence: 99%

Alphabet-Dependent Bounds for Linear Locally Repairable Codes Based on Residual Codes

Grezet

Freij-Hollanti

Westerbäck

et al. 2019

IEEE Trans. Inform. Theory

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“…The distance bound has been generalized in multiple ways, see e.g., [39], [48], [40], [49]. A large number of optimal code constructions have been presented, see e.g., [41], [50], [51], [9], [52], [53], [54].…”

Section: Related Work and Comparisonmentioning

confidence: 99%

Codes With Locality in the Rank and Subspace Metrics

Kadhe

Rouayheb

Duursma

et al. 2019

IEEE Trans. Inform. Theory

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We extend the notion of locality from the Hamming metric to the rank and subspace metrics. Our main contribution is to construct a class of array codes with locality constraints in the rank metric. Our motivation for constructing such codes stems from the need to design codes for efficient data recovery from correlated and/or mixed (i.e., complete and partial) failures in distributed storage systems. Specifically, the proposed local rank-metric codes can recover locally from crisscross errors and erasures, which affect a limited number of rows and/or columns of the storage array. We also derive a Singleton-like upper bound on the minimum rank distance of (linear) codes with rank-locality constraints. Our proposed construction achieves this bound for a broad range of parameters. The construction builds upon Tamo and Barg's method for constructing locally repairable codes with optimal minimum Hamming distance. Finally, we construct a class of constant-dimension subspace codes (also known as Grassmannian codes) with locality constraints in the subspace metric. The key idea is to show that a Grassmannian code with locality can be easily constructed from a rank-metric code with locality by using the lifting method proposed by Silva et al.We present an application of such codes for distributed storage systems, wherein nodes are connected over a network that can introduce errors and erasures.Index Terms-Codes for distributed storage, locally recoverable codes, rank-metric codes, subspace codes Server 2 Server 3 Server 4 Rack 2 X X X X X Server 1 X X X X X Server 2 Server 3 Server 4 Rack 1 X X X X X Server 1 X X X Server 2 Server 3 Server 4 Rack 3 X X Server 1 X X X X X X

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“…In [5], the puncture operation was defined to be the shorten operation here. It was claimed there the locality remains r, however as we shall see in the next example the shortened code can have locality r − 1.…”

Section: B Shorten Operationmentioning

confidence: 99%

“…By leveraging the results from the augment and extend operations, we conclude that if the minimum positive even weight of codewords in C ⊥ is w ⊥ min e , and the maximum Hamming weight of codewords in C ⊥ a is n + 1 − w ⊥ min e , then the lengthened code C ℓ has locality r ℓ = w ⊥ min e − 1. An operation similar to the lengthen operation, called enlarging, was proposed in [5]. Under the enlarging operation, the code length, dimension, and locality are increased by 1.…”

Section: Expurgate Augment and Lengthen Operationsmentioning

confidence: 99%

Cyclic linear binary locally repairable codes

Huang

Yaakobi

Uchikawa

et al. 2015

2015 IEEE Information Theory Workshop (ITW)

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Locally repairable codes (LRCs) are a class of codes designed for the local correction of erasures. They have received considerable attention in recent years due to their applications in distributed storage. Most existing results on LRCs do not explicitly take into consideration the field size q, i.e., the size of the code alphabet. In particular, for the binary case, only a few specific results are known by Goparaju and Calderbank. Recently, however, an upper bound on the dimension k of LRCs was presented by Cadambe and Mazumdar. The bound takes into account the length n, minimum distance d, locality r, and field size q, and it is applicable to both non-linear and linear codes.In this work, we first develop an improved version of the bound mentioned above for linear codes. We then focus on cyclic linear binary codes. By leveraging the cyclic structure, we notice that the locality of such a code is determined by the minimum distance of its dual code. Using this result, we investigate the locality of a variety of well known cyclic linear binary codes, e.g., Hamming codes and Simplex codes, and also prove their optimality with our improved bound for linear codes. We also discuss the locality of codes which are obtained by applying the operations of Extend, Shorten, Expurgate, Augment, and Lengthen to cyclic linear binary codes. Several families of such modified codes are considered and their optimality is addressed. Finally, we investigate the locality of Reed-Muller codes. Even though they are not cyclic, it is shown that some of the locality results for cyclic codes still apply.

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Constructions and Properties of Linear Locally Repairable Codes

Cited by 39 publications

References 25 publications

Alphabet-Dependent Bounds for Linear Locally Repairable Codes Based on Residual Codes

Alphabet-Dependent Bounds for Linear Locally Repairable Codes Based on Residual Codes

Codes With Locality in the Rank and Subspace Metrics

Cyclic linear binary locally repairable codes

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