On Optimal Quaternary Locally Repairable Codes

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.