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