An algorithm for counting arcs in higher-dimensional projective space

Abstract

An $n$ arc in $(k-1)$-dimensional projective space is a set of $n$ points so that no $k$ lie on a hyperplane. In 1988, Glynn gave a formula to count $n$-arcs in the projective plane in terms of simpler combinatorial objects called superfigurations. Several authors have used this formula to count $n$-arcs in the projective plane for $n\le 10$. In this paper, we determine a formula to count $n$-arcs in projective 3-space. We then use this formula to give exact expressions for the number of $n$-arcs in ${\mathbb{P}}^3({\mathbb{F}}_q)$ for $n\le 7$, which are polynomial in $q$ for $n\le 6$ and quasipolynomial in $q$ for $n=7$. Lastly, we generalize to higher-dimensional projective space.

arXiv commentRevised based on referee suggestions. Accepted to Finite Fields and Their Applications