Locally repairable codes (LRCs) are designed for distributed storage systems to reduce the repair bandwidth and disk I/O complexity during the storage node repair process. A code with $(r,\delta)$-locality (also called an $(r,\delta)$-LRC) can simultaneously repair up to $\delta-1$ symbols in a codeword by accessing at most $r$ other symbols in the codeword. In this paper, we propose a new method to calculate the $(r,\delta)$-locality of cyclic codes. Initially, we give a description of the algebraic structure of repeated-root cyclic codes of prime power lengths. Using this result, we derive a formula of $(r,\delta)$-locality of these cyclic codes for a wide range of $\delta$ values. Furthermore, we calculate the parameters of repeated-root cyclic codes of prime power lengths and obtain several infinite families of optimal cyclic $(r,\delta)$-LRCs, which exhibit new parameters compared with existing research on optimal $(r,\delta)$-LRCs with a cyclic structure. For the specific case of $\delta=2$, we have comprehensively identified all potential optimal cyclic $(r,2)$-LRCs of prime power lengths.

翻译：局部修复码（LRCs）是为分布式存储系统设计的，旨在降低存储节点修复过程中的修复带宽和磁盘I/O复杂度。具有$(r,\delta)$-局部性的码（也称为$(r,\delta)$-LRC）可通过访问码字中至多$r$个其他符号，同时修复该码字中最多$\delta-1$个符号。本文提出了一种计算循环码$(r,\delta)$-局部性的新方法。首先，我们描述了素数幂长度重根循环码的代数结构。基于此结果，对于广泛的$\delta$值范围，推导了这些循环码的$(r,\delta)$-局部性公式。此外，我们计算了素数幂长度重根循环码的参数，并获得了多个无限族最优循环$(r,\delta)$-LRCs，与现有关于循环结构最优$(r,\delta)$-LRCs的研究相比，这些码展示了新的参数。针对$\delta=2$的特殊情形，我们全面确定了所有可能的素数幂长度最优循环$(r,2)$-LRCs。