Cooperative Repair of Reed-Solomon Codes via Linearized Permutation Polynomials

In distributed storage, cooperative repair is to simultaneously recover
$h$
(
$h>1$
) node erasures by downloading data from surviving nodes as well as collaboration between the
$h$
replacement nodes. In this work, we propose a generalized cooperative repair framework for Reed-Solomon (RS) codes with two erasures. The key idea is to construct parity-check polynomials for the two replacement nodes respectively and then reduce the repair problem to the design of a linearized permutation polynomial related to the parity-check polynomials. We provide constructions of the linearized permutation polynomial in several cases, leading to cooperative repair schemes accordingly. Compared with the schemes given by Dau et al. 2021, our schemes retain the same repair bandwidth while apply to a much wider parameter regime and need only one-round collaboration. Finally we further reduce the repair bandwidth by the lifting method for RS codes of short length.