On the linear codes with (r, δ)-locality for distributed storage

Abstract

Recently linear codes with locality properties have attracted a lot of interest due to their desirable applications in distributed storage systems. An [n, k, d] linear code with (r, δ)-locality can enable the local recovery of a failed node in case of more than one node failures. In this paper, we study the theoretical bounds and constructions of linear codes with (r, δ)-locality for all code symbols. A parity-check matrix approach is employed to present an alternate simple proof of the Singleton-like bound for linear codes with all symbol (r, δ)-locality. A refined Singleton-like bound is given for the case that r | k and r + δ − 1 ∤ n. Base on the new proof technique, we enumerate all the possible two classes of optimal binary linear codes meeting the Singleton-like bound. In other words, except the proposed two classes of optimal binary linear codes, there is no other binary linear codes with minimum distance d = n — k — ([k/r] — 1)(δ− 1) + 1.