Binary cyclic codes that are locally repairable

Goparaju, Sreechakra; Calderbank, Robert

doi:10.1109/isit.2014.6874918

Cited by 109 publications

(114 citation statements)

References 23 publications

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“…We start with a simple result on the locality of the code symbols. Even though it has been mentioned before, see e.g., [6], [7], [13], [16], [17], we state and prove it here for completeness.…”

Section: Cyclic Linear Lrcsmentioning

confidence: 66%

“…Here we consider the [23,12,7] cyclic binary Golay code C. Its dual code C ⊥ is the [23,11,8] cyclic binary code. Hence, we conclude that C has locality r = 7 and the dual code C ⊥ has locality r ⊥ = 6.…”

Section: Cyclic Linear Lrcsmentioning

confidence: 99%

“…c d a r-optimality is proved in [2]. b k-optimality is proved for m > 8 in Theorem 3 of [7]. c With assumptions that the repair sets are disjoint and k is even, k is proved to achieve its upper bound in Theorem 2 of [7].…”

Section: Expurgate Augment and Lengthen Operationsmentioning

confidence: 99%

“…In this last construction the field size has to be at least the length of the code. In addition to the constructions mentioned above there are several other constructions of LRCs, see e.g., [4], [7]- [9], [12], [19].…”

Section: Introductionmentioning

confidence: 99%

“…The only construction that we know of for binary LRCs was presented recently by Goparaju and Calderbank [7].…”

Section: Introductionmentioning

confidence: 99%

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Cyclic linear binary locally repairable codes

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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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