A Construction of Optimal Quasi-cyclic Locally Recoverable Codes using Constituent Codes
Gustavo Terra Bastos, Angelynn Alvarez, Zachary Flores, Adriana Salerno
TL;DR
This paper addresses constructing quasi-cyclic locally recoverable codes (QC LRCs) with high rate and strong locality by decomposing QC codes into constituent codes over extension fields via CRT and by leveraging trace representations to connect to a baseline cyclic code D. It establishes a framework for analyzing minimum-distance performance through the Güneri-Özbudak bound $d_{GO}(C)$ and the Singleton-type bound $d_S(C)$, and introduces a constructive method to generate longer QC LRCs $C^j$ that preserve the constituent-code distances while reducing the Singleton bound gap. A central result, Theorem mainth, shows a descending chain of Singleton-type bounds under a simple condition, and proves that if a starting code is optimal then all expanded codes $C^j$ (for suitable $j$) remain optimal. The construction yields almost optimal or optimal QC LRCs with increased length and dimension, with potential applications to distributed storage systems and future avenues toward quantum error-correcting codes via constituent codes and self-orthogonality.
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.