Optimal cyclic (r,δ) locally repairable codes with unbounded length
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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. Lots of works have been proposed for constructions of optimal LRCs. Among these constructions, codes with large code length and small alphabet size are of special interests. Recently, Luo et al. 3 presented a construction of optimal r-LRCs of minimum distance 3 and 4 with unbounded lengths (i.e., lengths of these codes are independent of the alphabet size) 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.
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