🤖 AI Summary
This work investigates the weight distribution of repeated-root cyclic codes of prime-power length and their application to locally recoverable codes (LRCs), with emphasis on the algebraic characterization and optimal construction of $(r,delta)$-locality. Methodologically, it integrates algebraic coding theory, structural analysis of cyclic codes, polynomial factorization over finite fields, and derivation of locality bounds. The key contributions are: (i) the first explicit algebraic criterion for $(r,delta)$-locality in repeated-root cyclic codes; (ii) constructions of multiple infinite families of optimal cyclic $(r,delta)$-LRCs achieving the Singleton-type bound on minimum distance and locality; and (iii) a complete classification and enumeration of all optimal cyclic $(r,2)$-LRCs. The resulting optimal codes attain new parameter regimes—exhibiting unprecedented combinations of length, dimension, and locality—surpassing all previously known results in the literature.
📝 Abstract
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.