Algebraic constructions of complete $m$-arcs

Abstract

Let $m$ be a positive integer, $q$ be a prime power, and $\mathrm{PG}(2,q)$ be the projective plane over the finite field ${\mathbb{F}}_q$. Finding complete $m$-arcs in $\mathrm{PG}(2,q)$ of size less than $q$ is a classical problem in finite geometry. In this paper we give a complete answer to this problem when $q$ is relatively large compared with $m$, explicitly constructing the smallest $m$-arcs in the literature so far for any $m\ge 8$. For any fixed $m$, our arcs ${\mathcal{A}}_{q,m}$ satisfy $|{\mathcal{A}}_{q,m}|-q\to -\infty$ as $q$ grows. To produce such $m$-arcs, we develop a Galois theoretical machinery that allows the transfer of geometric information of points external to the arc, to arithmetic one, which in turn allows to prove the $m$-completeness of the arc.