Optimal Repair of MDS Codes in Distributed Storage via Subspace Interference Alignment

It is well known that an (n, k) code can be used to store information in a distributed storage system with n nodes/disks. If the storage capacity of each node/disk is normalized to one unit, the code can be used to store k units of information, where n > k. If the code used is maximum distance separable (MDS), then the storage system can tolerate up to (n−k) disk failures (erasures), since the original information can be reconstructed from any k surviving disks. The focus of this paper is the design of a systematic MDS code with the additional property that a single disk failure can be repaired with minimum repair bandwidth, i.e., with the minimum possible amount of data to be downloaded for recovery of the failed disk. Previously, a lower bound of n−1 n−k units has been established by Dimakis et. al, on the repair bandwidth for a single disk failure in an (n, k) MDS code based storage system, where each of the n disks store 1 unit of data. Recently, the existence of asymptotic codes achieving this lower bound for arbitrary (n, k) has been established by drawing connections to an asymptotic interference alignment scheme developed by Cadambe and Jafar for the interference channel. While the recent asymptotic constructions show the existence of codes achieving this lower bound in the limit of large code sizes, finite code constructions achieving this lower bound existed in previous literature only for the special (high-redundancy) scenario where k ≤ max(n/2, 3). The question of existence of finite codes for arbitrary values of (n, k) achieving the lower bound on the repair bandwidth remained open. As a main contribution of this paper, we provide the first known construction of a finite code for arbitrary (n, k), which can repair a single failed systematic disk by downloading exactly n−1 n−k units of data. The codes, which are optimally efficient in terms repair bandwidth are based on permutation matrices. We also show that our code has a simple repair property which enables efficiency, not only in terms of the amount of repair bandwidth, but also in terms of the amount of data accessed on the disk. We also generalize our permutation matrix based constructions by developing a novel framework for repair-bandwidth-optimal MDS codes based on the idea of subspace interference alignment a concept previously introduced by Suh and Tse the context of wireless cellular networks. This paper will be published, in part, in the Proceedings of IEEE Symposium on Information Theory (ISIT) 2011 [1]. 1The permutation marix based constructions of this paper have been discovered in parallel by Tamo et. al in [2]