On the construction of large local arcs

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.

Updated proofs leading to slightly larger local arcs in PG(2,q) for $q=p^m$ with m odd. Minor revisions to improve clarity and precision