# Optimal repair of Reed-Solomon codes: Achieving the cut-set bound

## TL;DR

This paper constructs Reed-Solomon codes that achieve the optimal repair bandwidth bound in distributed storage, solving a long-standing open problem by matching the cut-set bound with super-exponential sub-packetization.

## Contribution

The authors construct RS codes over large fields that meet the cut-set bound for repair bandwidth, and establish a lower bound on sub-packetization necessary for such optimal repair.

## Key findings

- RS codes constructed over fields with super-exponential size meet the cut-set bound.
- Lower bound on sub-packetization shows super-exponential growth is necessary.
- The work resolves the open problem of achieving optimal repair bandwidth with scalar MDS codes.

## Abstract

Coding for distributed storage gives rise to a new set of problems in coding theory related to the need of reducing inter-node communication in the system. A large number of recent papers addressed the problem of optimizing the total amount of information downloaded for repair of a single failed node (the repair bandwidth) by accessing information on $d$ {\em helper nodes}, where $k\le d\le n-1.$ By the so-called cut-set bound (Dimakis et al., 2010), the repair bandwidth of an $(n,k=n-r)$ MDS code using $d$ helper nodes is at least $dl/(d+1-k),$ where $l$ is the size of the node. Also, a number of known constructions of MDS array codes 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 these codes 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, 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 $q^l, l=\exp((1+o(1))n\log n)$ that meet the cut-set bound. We also prove an almost matching lower bound on $l$, showing that the 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 $l \ge \exp((1+o(1)) k\log k).$