Optimal ternary locally repairable codes

Jie Hao, Shu-Tao Xia, Kenneth W. Shum, Bin Chen, Fang-Wei Fu, Yixian Yang

Abstract: Locally repairable codes (LRCs) are linear codes with locality properties for code symbols, which have important applications in distributed storage systems. In this paper, we completely classify all the possible code parameters of optimal ternary LRCs achieving the Singleton-like bound proposed by Gopalan et al. Explicit constructions of optimal ternary LRCs are given for each group of possible code parameters. Moreover, it is also proved that optimal ternary LRCs with maximal minimum distance 6 are unique up to the equivalence of linear codes.