Codes With Locality for Two Erasures

Abstract

Codes with locality are a class of codes introduced by Gopalan <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">et al.</i> to efficiently repair a failed node, by minimizing the number of nodes contacted during repair. An <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$[n,k]$ </tex-math></inline-formula> systematic code is said to have information locality <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$r$ </tex-math></inline-formula> , if each message symbol can be recovered by accessing <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\leq r$ </tex-math></inline-formula> other symbols. An <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$[n,k]$ </tex-math></inline-formula> code is said to have all-symbol locality <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$r$ </tex-math></inline-formula> , if each code symbol can be recovered by accessing <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\leq r$ </tex-math></inline-formula> other symbols. In this paper, we consider a generalization of codes with all-symbol locality to the case of handling two erasures. We study codes with locality that can recover from two erasures via a sequence of two local, parity-check computations. We refer to these codes as sequential-recovery locally repairable codes (denoted by 2-seq LR codes). Earlier approaches to handling multiple erasures considered recovery in parallel; the sequential approach allows us to potentially construct codes with improved minimum distance. We derive an upper bound on the rate of 2-seq LR codes. We provide constructions based on regular graphs which are rate-optimal with respect to the derived bound. We also characterize the structure of any rate-optimal code. By studying the Generalized Hamming Weights of the dual code, we derive a recursive upper bound on the minimum distance of 2-seq LR codes. We also provide constructions of a family of codes based on Turán graphs, that are optimal with respect to this bound. We also present explicit distance-optimal Turán graph based constructions of 2-seq LR codes for certain parameters. Our approach also leads to a new bound on the minimum distance of codes with all-symbol locality for the single-erasure case.