Alphabet-Dependent Bounds for Linear Locally Repairable Codes Based on Residual Codes.

Locally repairable codes (LRC) have gained significant interest for the design of large distributed storage systems as they allow any small number of erased nodes to be recovered by accessing only a few others. Several works have thus been done to understand optimal rate-distance tradeoff and only recently the size of the alphabet has been taken into account. In this paper, a novel definition of locality is proposed to address the imprecision on the number of nodes contacted during the repair process. Then, a new alphabet-dependent bound is derived that applies to both definitions and shows better results than the currently known bounds. The bound is based on consecutive residual codes and intrinsically uses the Griesmer bound. Achievability results are also provided by considering the family of Simplex codes together with sporadic examples of optimal codes.