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.
Categories: math.CO