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.