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New Constructions of Optimal Cyclic (r,δ) Locally Repairable Codes from Their Zeros

An $(r, δ)$-locally repairable code ($(r, δ)$-LRC for short) was introduced by Prakash et al. \cite{Prakash2012} for tolerating multiple failed nodes in distributed storage systems, which was a generalization of the concept of $r$-LRCs produced by Gopalan et al. \cite{Gopalan2012}. An $(r, δ)$-LRC is said to be optimal if it achieves the Singleton-like bound. Recently, Chen et al. \cite{Chen2018} generalized the construction of cyclic $r$-LRCs proposed by Tamo et al. \cite{Tamo2015,Tamo2016} 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 \cite{Chen2018,Chen2019}, 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 \cite{Chen2018,Chen2019}, 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 $(r+δ-1)\nmid n$ can be obtained from our method.