On the Maximal Code Length of Optimal Linear Locally Repairable Codes

A code symbol in an $[n,\ k,\ d]$ linear code is said to have locality $r$ if it can be repaired from at most $r$ other code symbols. An $(n,\ k,\ r)$ locally repairable code (LRC) in which every code symbol has locality $r$ is said to be optimal if its minimum distance achieves the Singleton-like bound derived by Gopalan et al. In this paper, we study the maximal code length of a q-ary optimal $(n,\ k,\ r)$ -LRC. Firstly, we give an upper bound on the code length of q-ary optimal LRCs, and then derive some structural properties and the weight hierarchy of optimal LRCs with maximal code length. Finally, we give some constructions of optimal q-ary LRCs with maximal code length.