A Construction of Optimal Quasi-cyclic Locally Recoverable Codes using Constituent Codes

TL;DR

This paper introduces a method to construct optimal quasi-cyclic locally recoverable codes by decomposing them into constituent codes over field extensions, enhancing code parameters for distributed storage.

Contribution

It proposes a novel decomposition approach of quasi-cyclic LRCs into constituent codes, providing conditions for constructing almost optimal and optimal codes with larger dimensions and lengths.

Findings

Decomposition of quasi-cyclic LRCs into constituent codes over field extensions.

Conditions for the existence of almost optimal and optimal codes.

Enhanced code parameters with increased dimension and length.

Abstract

A locally recoverable code of locality $r$ over $\mathbb{F}_{q}$ is a code where every coordinate of a codeword can be recovered using the values of at most $r$ other coordinates of that codeword. Locally recoverable codes are efficient at restoring corrupted messages and data which make them highly applicable to distributed storage systems. Quasi-cyclic codes of length $n=m\ell$ and index $\ell$ are linear codes that are invariant under cyclic shifts by $\ell$ places. %Quasi-cyclic codes are generalizations of cyclic codes and are isomorphic to $\mathbb{F}_{q} [x]/ \langle x^m-1 \rangle$-submodules of $\mathbb{F}_{q^\ell} [x] / \langle x^m-1 \rangle$. In this paper, we decompose quasi-cyclic locally recoverable codes into a sum of constituent codes where each constituent code is a linear code over a field extension of $\mathbb{F}_q$. Using these constituent codes with set parameters, we propose conditions which ensure the existence of almost optimal and optimal quasi-cyclic locally recoverable codes with increased dimension and code length.