Extension of Optimal Locally Repairable codes

TL;DR

This paper introduces new constructions of optimal locally repairable codes (LRCs) using function fields, extending code lengths and providing novel methods for code extension while maintaining optimality.

Contribution

The paper presents several new constructions of optimal LRCs by extending existing codes and exploring elliptic and rational function fields, achieving longer code lengths and new optimal parameters.

Findings

Code length extended to q+2 over finite fields.

New construction of optimal (r,3)-LRCs by extending code blocks.

Optimal LRCs derived from elliptic function fields with lengths up to q+2√q-2r-2.

Abstract

Recent studies have delved into the construction of locally repairable codes (LRCs) with optimal minimum distance from function fields. In this paper, we present several novel constructions by extending the findings of optimally designed locally repairable codes documented in the literature. Let $C$ denote an optimal LRC of locality $r$, implying that every repairable block of $C$ is a $[r+1, r]$ MDS code, and $C$ maximizes its minimum distance. By extending a single coordinate of one of these blocks, we demonstrate that the resulting code remains an optimally designed locally repairable code. This suggests that the maximal length of an optimal LRC from rational function fields can be extended up to $q+2$ over a finite field $\mathbb{F}_q$. In addition, we give a new construction of optimal $(r, 3)$-LRC by extending one coordinate in each block within $C$. Furthermore, we propose a novel family of LRCs with Roth-Lempel type that are optimal under certain conditions. Finally, we explore optimal LRCs derived from elliptic function fields and extend a single coordinate of such codes. This approach leads us to confirm that the new codes are also optimal, thereby allowing their lengths to reach $q + 2\sqrt{q} - 2r - 2$ with locality $r$. We also consider the construction of optimal $(r, 3)$-LRC in elliptic function fields, with exploring one more condition.