Optimal Cyclic (r, ẟ) Locally Repairable Codes with Unbounded Length
Weijun Fang, Fang‐Wei Fu · 2018
Prakash et al. [2] introduced the concept of (r, δ) locally repairable codes ((r, δ)-LRCs for short) for tolerating multiple failed nodes. An (r, δ)-LRC is called optimal if it achieves the Singleton-type bound. In this paper, inspired by the work of [3], we firstly construct two classes of optimal cyclic (r, δ)-LRCs with unbounded lengths (i.e., lengths of these codes are independent of the alphabet size) 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 provide another class of optimal cyclic (r, 3)-LRCs with unbounded length and larger minimum distance 6.