Clay and Product-Matrix MSR Codes with Locality

Regenerating codes and codes with locality address the efficiency of the node repair problem. While regeneration reduces the repair bandwidth, locality reduces the number of nodes required for the repair of a small number of nodes. This work considers the combination of minimum storage regen-erating codes, an optimal subclass of regenerating codes, and PMDS codes, a particularly strong type of code with locality. Two new constructions are proposed in this paper. The first construction decreases the subpacketization mu n from rn to r r for all considered parameters of PMDS codes, and the second one achieves a significantly lower subpacketization n - r - 1 when the global redundancy is s = 1.