Constructions of Optimal $(r,\delta)$ Locally Repairable Codes via Constacyclic Codes

Abstract

Locally repairable codes (LRCs) are introduced in distributed storage systems due to their low repair overhead. An LRC is called optimal if its minimum distance attains the Singleton-like upper bound. Chen et al. (2018) recently studied the constructions of optimal $(r, \delta)$ -LRCs with length $n \mid (q+1)$ and $(r+\delta -1) \mid n$ , where many classes of optimal cyclic constructions were obtained. In this paper, by employing constacyclic MDS codes, we construct seven classes of optimal $(r, \delta)$ -LRCs with new parameters. After adding these new optimal LRCs via constacyclic codes, we have completely obtained all optimal $(r, \delta)$ -LRCs with length $n \mid (q+1)$ and $(r+\delta -1) \mid n$ for all possible parameters for the completeness in the coding theory. It is worth noting that the optimal constacyclic LRCs with new parameters provide more alternatives to cyclic LRCs in the practical demands of distributed storage systems, where specific values of $n$ , $k$ , $r$ , and $\delta $ are required. Moreover, constacyclic LRCs also possess the encoding and decoding efficiency as cyclic LRCs.