Repeated-root Constacyclic Codes with Optimal Locality

TL;DR

This paper explores repeated-root constacyclic codes, a generalization of cyclic codes, and constructs optimal locally repairable codes (LRCs) with efficient encoding, providing several infinite classes and proving optimality under specific conditions.

Contribution

It introduces the construction of optimal LRCs within repeated-root constacyclic codes and establishes their optimality over fixed alphabets, including classes over chain rings.

Findings

Several infinite classes of optimal LRCs are identified.

Optimal LRCs are constructed within the framework of repeated-root constacyclic codes.

Proves no other optimal LRC exists under the chain ring assumption.

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.