Repairing Reed-Solomon Codes Evaluated on Subspaces

TL;DR

This paper demonstrates the existence of efficient linear repair schemes for Reed-Solomon codes evaluated on subspaces, extending previous constructions to a broader parameter range with high success probability.

Contribution

It introduces new probabilistic repair schemes for Reed-Solomon codes on subspaces, applicable for both q≥3 and q=2, with explicit success probability bounds.

Findings

Existence of linear repair schemes with low bandwidth for RS codes on subspaces.

Extension of Dau--Milenkovich's construction to wider parameter ranges.

High success probability (at least 1/3) for the probabilistic repair schemes.

Abstract

We consider the repair problem for Reed--Solomon (RS) codes, evaluated on an $\mathbb{F}_q$-linear subspace $U\subseteq\mathbb{F}_{q^m}$ of dimension $d$, where $q$ is a prime power, $m$ is a positive integer, and $\mathbb{F}_q$ is the Galois field of size $q$. For the case of $q\geq 3$, we show the existence of a linear repair scheme for the RS code of length $n=q^d$ and codimension $q^s$, $s< d$, evaluated on $U$, in which each of the $n-1$ surviving nodes transmits only $r$ symbols of $\mathbb{F}_q$, provided that $ms\geq d(m-r)$. For the case of $q=2$, we prove a similar result, with some restrictions on the evaluation linear subspace $U$. Our proof is based on a probabilistic argument, however the result is not merely an existence result; the success probability is fairly large (at least $1/3$) and there is a simple criterion for checking the validity of the randomly chosen linear repair scheme. Our result extend the construction of Dau--Milenkovich to the range $r<m-s$, for a wide range of parameters.