Construction of Binary Locally Repairable Codes With Optimal Distance and Code Rate

Abstract

At present, the research on locally repairable codes (LRCs) in distributed storage systems indicates that, it is relatively easy to construct binary LRCs (BLRCs) with the optimal minimum distance, but under the condition of minimum distance upper bound, it remains difficult to construct the BLRCs with the optimal code rate. In order to solve the problem, BLRCs with optimal minimum distance and optimal code rate are constructed in this letter. The constructed linear codes are BLRCs with locality and availability of information symbols, and each repair group has only one parity symbol. Specifically, the method of block group design is adopted to construct the BLRCs with optimal minimum distance and optimal code rate of locality r=2 or availability t=2. Compared with the BLRCs proposed by W. Song et al, the BLRCs constructed in this letter perform better in code length and code rate. Furthermore, using the method of identity matrix transformation, BLRCs with optimal minimum distance and optimal code rate for availability t=2 are also constructed. Compared with the existing BLRCs based on graph construction, this construction has the same characteristics of minimum distance and code rate.