Nonlinear Repair of Reed-Solomon Codes

TL;DR

This paper introduces the first nonlinear repair scheme for Reed-Solomon codes that approaches optimal repair bandwidth, outperforming linear schemes and challenging existing bounds over prime fields.

Contribution

It presents the first nonlinear repair scheme for RS codes with asymptotically optimal bandwidth and demonstrates that the cut-set bound is not tight over prime fields.

Findings

First nonlinear repair scheme for RS codes with near-optimal bandwidth

Nonlinear schemes can outperform linear repair schemes

Tighter bounds for repair over prime fields using additive combinatorics

Abstract

The problem of repairing linear codes and, in particular, Reed Solomon (RS) codes has attracted a lot of attention in recent years due to their extreme importance to distributed storage systems. In this problem, a failed code symbol (node) needs to be repaired by downloading as little information as possible from a subset of the remaining nodes. By now, there are examples of RS codes that have efficient repair schemes, and some even attain the cut-set bound. However, these schemes fall short in several aspects; they require a considerable field extension degree. They do not provide any nontrivial repair scheme over prime fields. Lastly, they are all linear repairs, i.e., the computed functions are linear over the base field. Motivated by these and by a question raised in [GW17] on the power of nonlinear repair schemes, we study the problem of nonlinear repair schemes of RS codes. Our main results are the first nonlinear repair scheme of RS codes with asymptotically optimal repair bandwidth (asymptotically matching the cut-set bound). This is the first example of a nonlinear repair scheme of any code and also the first example that a nonlinear repair scheme can outperform all linear ones. Lastly, we show that the cut-set bound for RS codes is not tight over prime fields by proving a tighter bound, using additive combinatorics ideas.