On Optimal Quaternary Locally Repairable Codes

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

Abstract: A $q$-ary ($n, k, r$) locally repairable code (LRC) is an [$n, k, d$] linear code where every code symbol can be repaired by accessing at most $r$ other code symbols. Its minimum distance satisfies the well-known Singleton-like bound. In this paper, we determine all the possible parameters of quaternary LRCs attaining this Singleton-like bound by employing a parity-check matrix approach. Explicit optimal code constructions are given for all the possible parameters.