New Constructions of Optimal Cyclic (r,δ) Locally Repairable Codes from Their Zeros
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An (r, δ)-locally repairable code ((r, δ)-LRC for short) was introduced by Prakash et al. <cit.> for tolerating multiple failed nodes in distributed storage systems, which was a generalization of the concept of r-LRCs produced by Gopalan et al. <cit.>. An (r, δ)-LRC is said to be optimal if it achieves the Singleton-like bound. Recently, Chen et al. <cit.> generalized the construction of cyclic r-LRCs proposed by Tamo et al. <cit.> and constructed several classes of optimal (r, δ)-LRCs of length n for n | (q-1) or n | (q+1), respectively in terms of a union of the set of zeros controlling the minimum distance and the set of zeros ensuring the locality. Following the work of <cit.>, this paper first characterizes (r, δ)-locality of a cyclic code via its zeros. Then we construct several classes of optimal cyclic (r, δ)-LRCs of length n for n | (q-1) or n | (q+1), respectively from the product of two sets of zeros. Our constructions include all optimal cyclic (r,δ)-LRCs proposed in <cit.>, and our method seems more convenient to obtain optimal cyclic (r, δ)-LRCs with flexible parameters. Moreover, many optimal cyclic (r,δ)-LRCs of length n for n | (q-1) or n | (q+1), respectively such that (r+δ-1)∤ n can be obtained from our method.
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