On the construction of large local arcs

TL;DR

This paper introduces the concept of local arcs in finite projective geometry to construct large locally repairable codes with improved parameters, surpassing previous size limitations.

Contribution

It defines local arcs in PG(2,q), establishes bounds on their sizes, and constructs large examples leading to optimal LRCs with superlinear length.

Findings

Constructed k-uniform local arcs of size Ω(q^d) with d between 1.1167 and 1.25

Established bounds on maximum sizes of local arcs in PG(2,q)

Produced LRCs with length superlinear in q, improving previous results

Abstract

Motivated by the construction of optimal locally repairable codes, we introduce the new finite geometric concept of a \emph{local arc} which is defined as a collection $\mathcal{S}$ of disjoint point sets $S_{i}$ in $\mathrm{PG}(2,q)$ such that $S_{i} \cup S_{j}$ is an arc for any $S_{i}, S_{j} \in \mathcal{S}$. We focus on the upper and lower bounds on the sizes of maximum $k$-uniform local arcs. For $q=p^m$ with $p$ prime, we construct $k$-uniform local arcs in $\mathrm{PG}(2,q)$ of size $\Omega(q^{d})$ where $d$ is between $1.1167$ and $1.25$ depending only on $m$. For $k=4$, this implies the existence of optimal locally repairable codes (LRCs) with minimum distance 6, locality 3, and disjoint repair groups, whose length is superlinear in $q$--a significant improvement over the previously known $O(q)$ constructions for such LRCs.