Constructions of Locally Recoverable Codes which are Optimal

TL;DR

This paper introduces a Galois theoretical framework to construct optimal locally recoverable codes (LRCs), enabling the creation of new codes with improved parameters without relying on specific arithmetic properties.

Contribution

It develops a new Galois theoretical approach for constructing good polynomials, leading to the design of optimal LRCs with novel parameters and unifying existing theories.

Findings

Constructed new optimal LRCs with improved parameters

Developed a Galois theoretical framework for code construction

Unified existing theories of good polynomials

Abstract

Let $q$ be a prime power and $\mathbb F_q$ be the finite field of size $q$. In this paper we provide a Galois theoretical framework that allows to produce good polynomials for the Tamo and Barg construction of optimal locally recoverable codes (LRC). Using our approach we construct new good polynomials and then optimal LRCs with new parameters. The existing theory of good polynomials fits entirely in our new framework. The key advantage of our method is that we do not need to rely on arithmetic properties of the pair $(q,r)$, where $r$ is the locality of the code.