Locally repairable codes and minimal codes by homogenization of down-sets

In distributed storage systems, projective geometric self-repairing codes (PSRC) are able to provide a repair mechanism, which can repair a failed node with only two nodes. However, the redundancy of PSRC is extremely large, which makes PSRC unpractical for distributed storage systems. In this paper, we propose a new family of self-repairing codes called practical projective geometric self-repairing codes (PPSRC), which is a generalization of PSRC. PPSRC allow tradeoff between the coding rate and self-repairing capacity, while possess the basic property of PSRC. In addition, we provide an evolutionary repairing process for PPSRC, of which the repairing bandwidth is less than both the repairing process in [5] and that of the minimal storage repair codes (MSR) [3]. Furthermore, we present the general reconstruction process of PPSRC and the reconstruction process is optimal in terms of downloading bandwidth. Finally, we demonstrate PPSRC surpass the existing erasure coding strategies in terms of complexity, repairing nodes, repairing bandwidth and self-repairing capacity, which makes the PPSRC very attractive for distributed storage systems.