Some results on optimal locally repairable codes

Abstract

In a linear code, a code symbol is said to have locality r if it can be repaired by accessing at most r other code symbols. For an (n, k, r) locally repairable codes (LRC), the most important bounds on minimum distances might be the well-known Singleton-like bound and the Cadambe-Mazumdar bound which takes the field size into account. In this paper, we study the constructions of optimal LRCs from the view of parity-check matrices. Firstly, all the optimal binary LRCs meeting the Singleton-like bound are found in the sense of equivalence of linear codes, i.e., except the proposed 4 classes of LRCs, there is no other binary (n, k, r) LRC with minimum distance d = n − k − ⌈k/r⌉+2. Then a class of binary LRCs with distance 4 and arbitrary locality is proposed and shown to be optimal with respect to the Cadambe-Mazumdar bound. Moreover, we give a class of high rate optimal q-ary LRCs meeting the Singleton-like bound with minimum distance 4 while the required field size is only q ≥ r − 1. Finally, several methods to obtain short optimal LRCs from long optimal LRCs are proposed at the end of this paper.