The constructions of Singleton-optimal locally repairable codes with minimum distance 6 and locality 3

TL;DR

This paper introduces new constructions of optimal locally repairable codes with specific parameters using finite geometry, achieving flexible code lengths for prime power q ≥ 7.

Contribution

The paper provides a novel geometric approach to construct Singleton-optimal LRCs with minimum distance 6 and locality 3, expanding code length options based on properties of finite geometries.

Findings

Constructs codes with length 2q, 2q-2, or 2q-6 for prime power q ≥ 7.

Uses combinatorial structures from finite geometry, specifically arcs in projective planes.

Achieves Singleton-optimal LRCs with specified parameters.

Abstract

In this paper, we present new constructions of $q$-ary Singleton-optimal locally repairable codes (LRCs) with minimum distance $d=6$ and locality $r=3$, based on combinatorial structures from finite geometry. By exploiting the well-known correspondence between a complete set of mutually orthogonal Latin squares (MOLS) of order $q$ and the affine plane $\mathrm{AG}(2,q)$, We systematically construct families of disjoint 4-arcs in the projective plane $\mathrm{PG}(2,q)$, such that the union of any two distinct 4-arcs forms an 8-arc. These 4-arcs form what we call 4-local arcs, and their existence is equivalent to that of the desired codes. For any prime power $q\ge 7$, our construction yields codes of length $n = 2q$, $2q-2$, or $2q-6$ depending on whether $q$ is even, $q\equiv 3 \pmod{4}$, or $q\equiv 1 \pmod{4}$, respectively.