The (r, δ)-Locality of Repeated-Root Cyclic Codes with Prime Power Lengths

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.