The Repair Problem for Reed–Solomon Codes: Optimal Repair of Single and Multiple Erasures With Almost Optimal Node Size

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 <inline-formula> <tex-math notation="LaTeX">$h\geqslant 1$ </tex-math></inline-formula> failed nodes for an <inline-formula> <tex-math notation="LaTeX">$(n,k=n-r)$ </tex-math></inline-formula> maximum distance separable (MDS) code using <inline-formula> <tex-math notation="LaTeX">$d$ </tex-math></inline-formula> helper nodes is at least <inline-formula> <tex-math notation="LaTeX">$dhl/(d+h-k)$ </tex-math></inline-formula>, where <inline-formula> <tex-math notation="LaTeX">$l$ </tex-math></inline-formula> 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 the families of RS codes that achieve the cut-set bound for repair of one or several nodes. In the single-node case, we present the RS codes of length <inline-formula> <tex-math notation="LaTeX">$n$ </tex-math></inline-formula> over the field <inline-formula> <tex-math notation="LaTeX">${\mathbb F}_{q^{l}}, l=\exp ((1+o(1))n\log n)$ </tex-math></inline-formula> that meet the cut-set bound. We also prove an almost matching lower bound on <inline-formula> <tex-math notation="LaTeX">$l$ </tex-math></inline-formula>, 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 <inline-formula> <tex-math notation="LaTeX">$h=1,2, {\dots },r$ </tex-math></inline-formula> failed nodes from any subset of <inline-formula> <tex-math notation="LaTeX">$d$ </tex-math></inline-formula> helper nodes, <inline-formula> <tex-math notation="LaTeX">$k\leqslant d\leqslant n-h$ </tex-math></inline-formula>. For a fixed number of parities <inline-formula> <tex-math notation="LaTeX">$r$ </tex-math></inline-formula>, the node size of the constructed codes is close to the smallest possible node size for codes with such properties.