Algebraic approach to the completeness problem for $(k,n)$-arcs in planes over finite fields

TL;DR

This paper uses algebraic methods to analyze the completeness of certain small $(k,n)$-arcs derived from algebraic curves in finite projective planes, providing new criteria for their completeness based on Galois theory.

Contribution

It introduces an algebraic approach to determine the completeness of $(k,n)$-arcs from Hermitian and BKS curves, advancing understanding of their structure and properties in finite geometry.

Findings

Hermitian curve-based arcs are complete for $r \\ge 4$.

BKS curve-based arcs are complete if and only if $r$ is even.

Adding uncovered points yields larger complete arcs in certain cases.

Abstract

In a projective plane over a finite field, complete $(k,n)$-arcs with few characters are rare but interesting objects with several applications to finite geometry and coding theory. Since almost all known examples are large, the construction of small ones, with $k$ close to the order of the plane, is considered a hard problem. A natural candidate to be a small $(k,n)$-arc with few characters is the set $\Omega(\mathcal{C})$ of the points of a plane curve $\mathcal{C}$ of degree $n$ (containing no linear components) such that some line meets $\mathcal{C}$ transversally in the plane, i.e. in $n$ pairwise distinct points. Let $\mathcal{C}$ be either the Hermitian curve of degree $q+1$ in $\mathrm{PG}(2,q^{2r})$ with $r\ge 1$, or the rational BKS curve of degree $q+1$ in $\mathrm{PG}(2,q^r)$ with $q$ odd and $r\ge 1$. Then $\Omega(\mathcal{C})$ has four and seven characters, respectively. Furthermore, $\Omega(\mathcal{C})$ is small as both curves are either maximal or minimal. The completeness problem is investigated by an algebraic approach based on Galois theory and on the Hasse-Weil lower bound. Our main result for the Hermitian case is that $\Omega(\mathcal{C})$ is complete for $r\ge 4$. For the rational BKS curve, $\Omega(\mathcal{C})$ is complete if and only if $r$ is even. If $r$ is odd then the uncovered points by the $(q+1)$-secants to $\Omega(\mathcal{C})$ are exactly the points in $\mathrm{PG}(2,q)$ not lying in $\Omega(\mathcal{C})$. Adding those points to $\Omega(\mathcal{C})$ produces a complete $(k,q+1)$-arc in $\mathrm{PG}(2,q^r)$, with $k=q^r+q$. The above results do not hold true for $r=2$ and there remain open the case $r=3$ for the Hermitian curve, and the cases $r=3,4$ for the rational BKS curve. As a by product we also obtain two results of interest in the study of the Galois inverse problem for $\mathrm{PGL}(2,q)$.