The repair problem for Reed-Solomon codes: Optimal repair of single and multiple erasures, asymptotically optimal node size
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The repair problem in distributed storage addresses recovery of the data encoded using an erasure code, for instance, a Reed-Solomon (RS) code. We consider the problem of repairing a single node or multiple nodes in RS-coded storage systems using the smallest possible amount of inter-nodal communication. According to the cut-set bound, communication cost of repairing h> 1 failed nodes for an (n,k=n-r) MDS code using d helper nodes is at least dhl/(d+h-k), where l is the size of the node. Guruswami and Wootters (2016) initiated the study of efficient repair of RS codes, showing that they can be repaired 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. In this paper we construct families of RS codes that achieve the cutset bound for repair of one or several nodes. In the single-node case, we present RS codes of length n over the field F_q^l,l=((1+o(1))n 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 scalar MDS codes to achieve the cut-set bound using linear repair schemes. For the case of multiple nodes, we construct a family of RS codes that achieve the cut-set bound universally for the repair of any h=2,3,... failed nodes from any subset of d helper nodes, k< d< n-h. For a fixed number of parities r the node size of the constructed codes is close to the smallest possible node size for codes with such properties.
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