Optimal linear codes with a local-error-correction property

Abstract

Motivated by applications to distributed storage, Gopalan et al recently introduced the interesting notion of information-symbol locality in a linear code. By this it is meant that each message symbol appears in a parity-check equation associated with small Hamming weight, thereby enabling recovery of the message symbol by examining a small number of other code symbols. This notion is expanded to the case when all code symbols, not just the message symbols, are covered by such "local" parity. In this paper, we extend the results of Gopalan et. al. so as to permit recovery of an erased code symbol even in the presence of errors in local parity symbols. We present tight bounds on the minimum distance of such codes and exhibit codes that are optimal with respect to the local error-correction property. As a corollary, we obtain an upper bound on the minimum distance of a concatenated code. ] linear code C over the field F q is said to have locality r if this symbol can be recovered by accessing at most r other code symbols of code C. Equivalently, for any coordinate i, there exists a row in the parity-check matrix of the code of Hamming weight at most r + 1, whose support includes i. An (r, d) code was defined as a systematic linear code C having minimum distance d, where all k message symbols have locality r. It was shown that the minimum distance of an (r, d) code is upper bounded by

Optimal locally repairable codes with local minimum storage regeneration via rank-metric codes

“…The paper also showes that pyramid codes, presented in [17], achieve this bound with information symbols locality. Subsequently, the work by Prakash et al extends the bound to a more general definition of scalar LRCs [11]. (Han and Lastras-Montano [18] provide a similar upper bound which is coincident with the one in [11] for small minimum distances, and also present codes that attain this bound in the context of reliable memories.)…”

“…Subsequently, the work by Prakash et al extends the bound to a more general definition of scalar LRCs [11]. (Han and Lastras-Montano [18] provide a similar upper bound which is coincident with the one in [11] for small minimum distances, and also present codes that attain this bound in the context of reliable memories.) In [10], Papailiopoulos and Dimakis generalize the bound in [9] to vector codes, and present locally repairable coding schemes which exhibits MDS property at the cost of small amount of additional storage per node.…”

“…In [11], Prakash et al present the following upper bound on the minimum distance of an (r, δ) scalar LRC:…”

“…It was established in [11] that a family of pyramid codes, presented in [17], attains this bound and has information locality, i.e. only information symbols satisfy the locality constraint.…”

“…However, an explicit construction of optimal scalar LRCs with all-symbols locality is known only for the case n = M r (r + δ − 1) [11], [18]. Towards optimal scalar LRCs for broader range of parameters, given field size |F| > Mn M , [11] establishes the existence of scalar codes with all-symbols locality for the setting when (r + δ − 1)|n. In this paper, we provide an explicit construction of optimal scalar LRCs with all-symbols locality relaxing the restriction of (r + δ − 1)|n.…”