In a distributed storage system, code symbols are dispersed across space in nodes or storage units as opposed to time. In settings such as that of a large data center, an important consideration is the efficient repair of a failed node. Efficient repair calls for erasure codes that in the face of node failure, are efficient in terms of minimizing the amount of repair data transferred over the network, the amount of data accessed at a helper node as well as the number of helper nodes contacted. Coding theory has evolved to handle these challenges by introducing two new classes of erasure codes, namely regenerating codes and locally recoverable codes as well as by coming up with novel ways to repair the ubiquitous Reed-Solomon code. This survey provides an overview of the efforts in this direction that have taken place over the past decade.
In @cite_182 , the authors improve the Guruswami-Wootters approach to a larger class of parameters. In @cite_87 , the authors provide a family of RS codes that has asymptotically optimal repair bandwidth with respect to the cut-set bound. This result is further developed in @cite_184 to reduce the sub-packetization levels. In @cite_112 , the authors present RS codes that meet the MSR point for all parameters: @math . Bandwidth-efficient recovery from multiple erasures in RS codes is addressed in @cite_153 and is further extended to include general scalar MDS codes in @cite_145 . In @cite_152 , the authors present codes that universally achieve the optimal bandwidth points for all parameters @math and @math simultaneously.
@cite_87 @cite_152 @cite_184 @cite_145 @cite_112 @cite_153 @cite_182
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The repair problem for an (n, k) error-correcting code calls for recovery of an unavailable coordinate of the codeword by downloading as little information as possible from a subset of the remaining coordinates. Using the terminology motivated by coding in distributed storage, we attempt to repair a failed node by accessing information stored on d helper nodes, where k &#x227C; d &#x227C; n &#x2013; 1, and using as little repair bandwidth as possible to recover the lost information.By the so-called cut-set bound (, 2010), the repair bandwidth of an (n,k = n &#x2013; r) MDS code using d helper nodes is at least dl (d + 1 &#x2013; k), where l is the size of the node. A number of constructions of MDS array codes have been shown to meet this bound with equality. In a related but separate line of work, Guruswami and Wootters (2016) studied repair of Reed-Solomon (RS) codes, showing that it is possible to perform repair using a smaller bandwidth than under the trivial approach. At the same time, their work as well as follow-up papers stopped short of constructing RS codes (or any scalar MDS codes) that meet the cut-set bound with equality, which has been an open problem in coding theory.In this work we present a solution to this problem, constructing RS codes of length n over the field of size (ql, l = exp((1 + o(1)n log n) that meet the cut-set bound. We also prove an almost matching lower bound on l, showing that super-exponential scaling is both necessary and sufficient for achieving the cut-set bound using linear repair schemes. More precisely, we prove that for scalar MDS codes (including the RS codes) to meet this bound, the sub-packetization l must satisfy l &#x227D; exp((1 + o(1))k log k). 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. The repair bandwidth of a code is the minimum amount of data required to repair one or several failed nodes (erasures). For MDS codes, the repair bandwidth is bounded below by the so-called cut-set bound, and codes that meet this bound with equality are said to support optimal repair of one or multiple failed nodes. We consider the problem of repairing multiple failed nodes of Reed-Solomon (RS) codes. In a recent work with I. Tamo (Proc. IEEE FOCS 2017), we gave the first explicit construction of RS codes with optimal repair of any single failed node from any subset of helper nodes. In this paper, we construct explicit RS codes that universally achieve the cut-set bound for the repair of any number of failed nodes from any set of helper nodes. Moreover, the node size of our codes is close to the optimal (smallest possible) node size of codes with such property. The high repair bandwidth cost of (n,k) maximum distance separable (MDS) erasure codes has motivated a new class of codes that can reduce repair bandwidth over that of conventional MDS codes. In this paper, we address (n,k,d) exact repair MDS codes, which allow for any single failed node to be repaired exactly with access to any arbitrary set of d survivor nodes. We show the existence of exact repair MDS codes that achieve minimum repair bandwidth (matching the cut-set lower bound) for arbitrary admissible (n,k,d), i.e., k ≤ d ≤ n-1. Moreover, we extend our results to show the optimality of our codes for multiple-node failure scenarios in which an arbitrary set of r ≤ n-k failed nodes needs to repaired. Our approach is based on asymptotic interference alignment proposed by Cadambe and Jafar. As a byproduct, we also characterize the capacity of a class of multisource nonmulticast networks.