Optimal cyclic (r,δ) locally repairable codes with unbounded length

Abstract

Locally repairable codes with locality r (r-LRCs for short) were introduced by Gopalan et al. [1] to recover a failed node of the code from at most other r available nodes. And then (r,δ)-locally repairable codes ((r,δ)-LRCs for short) were produced by Prakash et al. [2] for tolerating multiple failed nodes. An r-LRC can be viewed as an (r,2)-LRC. An (r,δ)-LRC is called optimal if it achieves the Singleton-type bound. It has been a great challenge to construct q-ary optimal (r,δ)-LRCs with length much larger than q. Surprisingly, Luo et al. [3] presented a construction of q-ary optimal r-LRCs of minimum distances 3 and 4 with unbounded lengths (i.e., lengths of these codes are independent of q) via cyclic codes.In this paper, inspired by the work of [3], we firstly construct two classes of optimal cyclic (r,δ)-LRCs with unbounded lengths and minimum distances δ+1 or δ+2, which generalize the results about the δ=2 case given in [3]. Secondly, with a slightly stronger condition, we present a construction of optimal cyclic (r,δ)-LRCs with unbounded length and larger minimum distance 2δ. Furthermore, when δ=3, we give another class of optimal cyclic (r,3)-LRCs with unbounded length and minimum distance 6.