Repeated-root Constacyclic Codes with Optimal Locality
Abstract
A code is called a locally repairable code (LRC) if any code symbol is a function of a small fraction of other code symbols. When a locally repairable code is employed in a distributed storage systems, an erased symbol can be recovered by accessing only a small number of other symbols, and hence alleviating the network resources required during the repair process. In this paper we consider repeated-root constacyclic codes, which is a generalization of cyclic codes, that are optimal with respect to a Singleton-like bound on minimum distance. An LRC with the structure of a constacyclic code can be encoded efficiently using any encoding algorithm for constacyclic codes in general. In this paper we obtain optimal LRCs among these repeated-root constacyclic codes. Several infinite classes of optimal LRCs over a fixed alphabet are found. Under a further assumption that the ambient space of the repeated-root constacyclic codes is a chain ring, we show that there is no other optimal LRC.
Cite
@article{arxiv.2111.12489,
title = {Repeated-root Constacyclic Codes with Optimal Locality},
author = {Wei Zhao and Kenneth W. Shum and Shenghao Yang},
journal= {arXiv preprint arXiv:2111.12489},
year = {2021}
}