An $(r, \delta)$-locally repairable code ($(r, \delta)$-LRC for short) was introduced by Prakash et al. for tolerating multiple failed nodes in distributed storage systems, and has garnered significant interest among researchers. An $(r,\delta)$-LRC is called an optimal code if its parameters achieve the Singleton-like bound. In this paper, we construct three classes of $q$-ary optimal cyclic $(r,\delta)$-LRCs with new parameters by investigating the defining sets of cyclic codes. Our results generalize the related work of \cite{Chen2022,Qian2020}, and the obtained optimal cyclic $(r, \delta)$-LRCs have flexible parameters. A lot of numerical examples of optimal cyclic $(r, \delta)$-LRCs are given to show that our constructions are capable of generating new optimal cyclic $(r, \delta)$-LRCs.

翻译：Prakash等人针对分布式存储系统中容忍多节点故障问题提出了$(r,\delta)$-局部可修复码（简称$(r,\delta)$-LRC），该编码已引起研究者的广泛关注。若$(r,\delta)$-LRC的参数达到Singleton类界，则称为最优码。本文通过研究循环码的定义集，构造了三类具有新参数的$q$-元最优循环$(r,\delta)$-LRC。我们的结果推广了\cite{Chen2022,Qian2020}的相关工作，且得到的最优循环$(r,\delta)$-LRC具有灵活的参数。通过大量数值实例展示了我们的构造方法能够生成新型最优循环$(r,\delta)$-LRC。