Upper bounds on the smallest size of a complete cap in $\mathrm{PG}(N,q)$, $N\ge3$, under a certain probabilistic conjecture

TL;DR

This paper derives new probabilistic upper bounds on the smallest size of complete caps in projective spaces over finite fields, based on a conjecture about the distribution of uncovered points.

Contribution

It introduces a probabilistic conjecture and rigorously proves bounds for part of the iterative construction of complete caps in PG(N,q).

Findings

New upper bounds on the size of complete caps in PG(N,q) for N≥3.

Connection established between cap sizes and the Birthday problem.

Bounds validated through comparison with computational data.

Abstract

In the projective space $\mathrm{PG}(N,q)$ over the Galois field of order $q$, $N\ge3$, an iterative step-by-step construction of complete caps by adding a new point on every step is considered. It is proved that uncovered points are evenly placed on the space. A natural conjecture on an estimate of the number of new covered points on every step is done. For a part of the iterative process, this estimate is proved rigorously. Under the conjecture mentioned, new upper bounds on the smallest size $t_{2}(N,q)$ of a complete cap in $\mathrm{PG}(N,q)$ are obtained, in particular, \begin{align*} t_{2}(N,q)<\frac{\sqrt{q^{N+1}}}{q-1}\left(\sqrt{(N+1)\ln q}+1\right)+2\thicksim q^\frac{N-1}{2}\sqrt{(N+1)\ln q},\quad N\ge3. \end{align*} A connection with the Birthday problem is noted. The effectiveness of the new bounds is illustrated by comparison with sizes of complete caps obtained by computer in wide regions of $q$.