Improved schemes for asymptotically optimal repair of MDS codes

Abstract

An (n, k, l) MDS code of length n, dimension k and sub-packetization l over a finite field F is a set of n symbol vectors of length l over F with the property that any k vector can recover the entire data of kl symbols. When a node fails we can recover it by downloading symbols from the surviving nodes, and the total number of symbols downloaded in th worst case is the repair bandwidth of the code. By the cut-se bound, the repair bandwidth of an (n, k, l) MDS code is at leas (n − 1)l/(n − k). There are several constructions of (n, k, l) MDS codes whose repair bandwidths meet or asymptotically meet the cut-se bound. Letting r = n − k denote the number of parities Ye and Barg constructed (n, k, rn) Reed-Solomon codes the asymptotically meet the cut-set bound. Ye and Barg also constructed optimal bandwidth and optimal update (n, k, rn MDS codes. A key idea in these constructions is to expand integers in base r. We show in this paper that, when r is an integral power, w can significantly reduce the sub-packetization of the Ye-Barg constructions while achieving asymptotically optimal repair bandwidth. As an example, when r = 2m, we achieve the sub-packetization of 2m+n−1, which improves upon the sub packetization of 2mn in the Ye-Barg constructions. In general when r = sm for an integer s  2, our codes have sub packetization l = sm+n−1 = rsn−1. Specifically, in the casi r = sm, we obtain an (n, k, sm+n=1) Reed-Solomon code an an optimal update (n, k, sm+n−1) MDS code, which both have asymptotically optimal repair bandwidth. In order to obtain these results, we extend and generalize the r-ary expansion idea in the Ye-Barg constructions. Even when r is not an integral power, we can still obtain (n, k, sm+n−1) Reed-Solomon codes and optimal update (n, k, sm+n−1) MDS codes by choosing positive integers s an m such that sm  r. In this case, however, the resulting codes have bandwidth that is near-optimal rather than asymptotically optimal.