Explicit Constructions of Optimal-Access MDS Codes With Nearly Optimal Sub-Packetization

Abstract

An (n, k, l) maximum distance separable (MDS) array code of length n, dimension k = n - r, and subpacketization l is formed of l × n matrices over a finite field F, with every column of the matrix stored on a separate node in the distributed storage system and viewed as a coordinate of the codeword. Repair of a failed node (recovery of one erased column) can be performed by accessing a set of d ≤ n - 1 surviving (helper) nodes. The code is said to have the optimal access property if the amount of data accessed at each of the helper nodes meets a lower bound on this quantity. For optimal-access MDS codes with d = n-1, the sub-packetization l satisfies the bound l ≥ r <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">(k-1)/r</sup> . In our previous work (IEEE Trans. Inf. Theory, vol. 63, no. 4, 2017), for any n and r, we presented an explicit construction of optimal-access MDS codes with subpacketization l = r <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">⌈n-1⌉</sup> . In this paper, we take up the question of reducing the sub-packetization value l to make it to approach the lower bound. We construct an explicit family of optimal-access codes with l = r1.n/rl, which differs from the optimal value by at most a factor of r <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> . These codes can be constructed over any finite field F as long as |F| ≥ r⌈n/r⌉, and afford low-complexity encoding and decoding procedures. We also define a version of the repair problem that bridges the context of regenerating codes and codes with locality constraints (LRC codes), which we call group repair with optimal access. In this variation, we assume that the set of n = sm nodes is partitioned into m repair groups of size s, and require that the amount of accessed data for repair is the smallest possible whenever the d = s + k - 1 helper nodes include all the other s-1 nodes from the same group as the failed node. For this problem, we construct a family of codes with the group optimal access property. These codes can be constructed over any field F of size |F| ≥ n, and also afford low-complexity encoding and decoding procedures.