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 $q > 2$, 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 $q=2$, 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.