Minimum Storage Regenerating Codes for All Parameters

Abstract

Abstract-Regenerating codes for distributed storage have attracted much research interest in the past decade. Such codes trade the bandwidth needed to repair a failed node with the overall amount of data stored in the network. Minimum storage regenerating (MSR) codes are an important class of optimal regenerating codes that minimize (first) the amount of data stored per node and (then) the repair bandwidth. Specifically, an [n, k, d]-(α) MSR code C over F q stores a file F consisting of αk symbols over F q among n nodes, each storing α symbols, in such a way that:• the file F can be recovered by downloading the content of any k of the n nodes; and • the content of any failed node can be reconstructed by accessing any d of the remaining n − 1 nodes and downloading α/(d−k+1) symbols from each of these nodes. In practice, the file F is typically available in uncoded form on some k of the n nodes, known as systematic nodes, and the defining node-repair condition above can be relaxed to requiring the optimal repair bandwidth for systematic nodes only. Such codes are called systematic-repair MSR codes.Unfortunately, finite-α constructions of [n, k, d] MSR codes are known only for certain special cases: either low rate, namely k/n 0.5, or high repair connectivity, namely d = n − 1. Our main result in this paper is a finite-α construction of systematic-repair [n, k, d] MSR codes for all possible values of parameters n, k, d. We also introduce a generalized construction for [n, k] MSR codes to achieve the optimal repair bandwidth for all values of d simultaneously.

Systematic Construction of MDS Codes with Small Sub-packetization Level and Near Optimal Repair Bandwidth

“…Similarly to Theorem 7 in [10] and Lemma 7 in [15], by virtue of Lemma 2 we have the following result to ensure the non-singularity of the related sub-block matrices. Proof.…”

“…Theorem 3. Choosing the Ye-Barg code 1, defined by the coding matrices (15), as the base code for the generic transformation in Section III, we can get an (n = sn ′ , k) MDS code C 1 over F q with q > N n−1 r−1 , where k = n − r. Specifically, the subpacketization level of the MDS code C 1 is r n ′ while the repair bandwidth is (1 + (s−1)(r−1) n−1 ) times the optimal value. B.…”

Asymptotically Optimal Regenerating Codes Over Any Field

“…In particular, the problem of constructing high rate MSR codes, i.e., with a constant number of parity nodes, has received a great deal of attention [4,22,24,29,30]. Implementing our techniques for high rate MSR codes is one of our future research directions.…”

MDS Code Constructions with Small Sub-packetization and Near-optimal Repair Bandwidth

“…Towards constructing high-rate exact-repairable MDS codes with optimal repair-bandwidth, Cadambe et al [3] show the existence of such codes when subpacketization level approaches infinity. Motivated by this result, the problem of designing high-rate exactrepairable MDS codes with finite sub-packetization level and optimal repair bandwidth is explored in [2,8,17,22,23,28,32,34,36] and references therein.…”

“…The problem of constructing MDS codes with optimal repair-bandwidth (cf. (1.1)) has been explored in [3,8,17,18,22,23,28,36] and references therein. Note that as the number of the nodes contacted during the repair process d gets larger, the optimal repair bandwidth defined by the cut-set bound becomes significantly smaller than the naive repair bandwidth of k symbols (over F) or k symbols (over B).…”