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

TL;DR

New constructions of $q-ary Singleton-optimal locally repairable codes (LRCs) with minimum distance $d=6$ and locality $r=3$ are presented, based on combinatorial structures from finite geometry.

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.

Figures (1)

• Figure 1: A 4-local arc constructed from a Latin square of order $8$.

Theorems & Definitions (19)

• Lemma 1.1
• Theorem 1.2
• Lemma 2.1
• Definition 2.2: 4-local arc
• Remark 2.1
• Lemma 3.1
• Remark 3.1
• Proposition 3.2
• Lemma 4.1
• Theorem 4.2