Construction of optimal locally repairable codes via automorphism groups of rational function fields

TL;DR

This paper introduces a novel method for constructing optimal locally repairable codes using automorphism groups of rational function fields, achieving codes with desirable properties and smaller alphabet sizes.

Contribution

It presents a new algebraic approach employing automorphism groups to construct optimal locally repairable codes with improved parameters.

Findings

Constructed new families of q-ary locally repairable codes

Achieved codes with smaller alphabet sizes compared to code length

Produced codes of length q+1 via cyclic and dihedral groups

Abstract

Locally repairable codes, or locally recoverable codes (LRC for short) are designed for application in distributed and cloud storage systems. Similar to classical block codes, there is an important bound called the Singleton-type bound for locally repairable codes. In this paper, an optimal locally repairable code refers to a block code achieving this Singleton-type bound. Like classical MDS codes, optimal locally repairable codes carry some very nice combinatorial structures. Since introduction of the Singleton-type bound for locally repairable codes, people have put tremendous effort on constructions of optimal locally repairable codes. Due to hardness of this problem, there are few constructions of optimal locally repairable codes in literature. Most of these constructions are realized via either combinatorial or algebraic structures. In this paper, we employ automorphism groups of rational function fields to construct optimal locally repairable codes by considering the group action on the projective lines over finite fields. It turns out that we are able to construct optimal locally repairable codes with reflexibility of locality as well as smaller alphabet size comparable to the code length. In particular, we produce new families of $q$-ary locally repairable codes, including codes of length $q+1$ via cyclic groups and codes via dihedral groups.