On the Maximal Code Length of Optimal Linear Locally Repairable Codes
Jie Hao, Kenneth W. Shum, Shu‐Tao Xia, Yixian Yang · 2018
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.