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An $(r, \delta)$-locally repairable code ($(r, \delta)$-LRC for short) wasintroduced by Prakash et al. \cite{Prakash2012} for tolerating multiple failednodes in distributed storage systems, which was a generalization of the conceptof $r$-LRCs produced by Gopalan et al. \cite{Gopalan2012}. An $(r, \delta)$-LRCis said to be optimal if it achieves the Singleton-like bound. Recently, Chenet al. \cite{Chen2018} generalized the construction of cyclic $r$-LRCs proposedby Tamo et al. \cite{Tamo2015,Tamo2016} and constructed several classes ofoptimal $(r, \delta)$-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 minimumdistance and the set of zeros ensuring the locality. Following the work of\cite{Chen2018,Chen2019}, this paper first characterizes $(r, \delta)$-localityof a cyclic code via its zeros. Then we construct several classes of optimalcyclic $(r, \delta)$-LRCs of length $n$ for $n\, |\, (q-1)$ or $n\,|\, (q+1)$,respectively from the product of two sets of zeros. Our constructions includeall optimal cyclic $(r,\delta)$-LRCs proposed in \cite{Chen2018,Chen2019}, andour method seems more convenient to obtain optimal cyclic $(r, \delta)$-LRCswith flexible parameters. Moreover, many optimal cyclic $(r,\delta)$-LRCs oflength $n$ for $n\, |\, (q-1)$ or $n\,|\, (q+1)$, respectively such that$(r+\delta-1)\nmid n$ can be obtained from our method.

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