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

TL;DR

This paper introduces an algorithm to count n-caps in three-dimensional projective space over finite fields, providing exact formulas for n up to 7, with polynomial and quasipolynomial expressions depending on n.

Contribution

It presents a new algorithm for counting n-caps in P^3(F_q) and derives explicit formulas for n up to 7, including polynomial and quasipolynomial cases.

Findings

Exact formulas for n-caps when n ≤ 7.

Polynomial formulas for n ≤ 6.

Quasipolynomial formula for n = 7.

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 $n = 7$.