Repairing Reed-Solomon Codes Evaluated on Subspaces

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 $q\ge 3$, 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 $ms\ge d(m-r)$. 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 $1\mathrm{/}3$) 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.