An algorithm to count the number of caps in ${\mathbb{P}}^3({\mathbb{F}}_q)$

Abstract

An $n$-cap in $k$-dimensional projective space is a set of $n$ points so that no three lie on a line. In this note, we provide an algorithm to count the number of $n$-caps in ${\mathbb{P}}^3({\mathbb{F}}_q)$, which follows from our recent paper [9]. We then give exact formulas for the number of $n$-caps when $n\le 7$. The formulas are polynomial in $q$ when $n\le 6$ and quasipolynomial in $q$ when $n=7$.

arXiv comment8 pages