Tables, bounds and graphics of the smallest known sizes of complete caps in the spaces $\mathrm{PG}(3,q)$ and $\mathrm{PG}(4,q)$

TL;DR

This paper presents computational results on small complete caps in projective spaces PG(3,q) and PG(4,q), establishing new smallest known sizes and deriving improved upper bounds for the minimum size of complete caps.

Contribution

The study introduces new smallest known complete caps in PG(3,q) and PG(4,q) over wide q regions using randomized algorithms, leading to new upper bounds on t_2(N,q).

Findings

New smallest complete caps found for PG(3,q) and PG(4,q).

Derived new upper bounds on minimum complete cap sizes.

Results suggest these bounds may hold universally for all q.

Abstract

In this paper we present and analyze computational results concerning small complete caps in the projective spaces $\mathrm{PG}(N,q)$ of dimension $N=3$ and $N=4$ over the finite field of order $q$. The results have been obtained using randomized greedy algorithms and the algorithm with fixed order of points (FOP). The computations have been done in relatively wide regions of $q$ values; such wide regions are not considered in literature for $N=3,4$. The new complete caps are the smallest known. Basing on them, we obtained new upper bounds on $t_2(N,q)$, the minimum size of a complete cap in $\mathrm{PG}(N,q)$, in particular, \begin{align*} &t_{2}(N,q)<\sqrt{N+2}\cdot q^{\frac{N-1}{2^{\vphantom{H}}}}\sqrt{\ln q},\quad q\in L_{N},\quad N=3,4,\\ &t_{2}(N,q)<\left(\sqrt{N+1}+\frac{1.3}{\ln (2q)}\right)q^{\frac{N-1}{2^{\vphantom{H}}}}\sqrt{\ln q},\quad q\in L_{N},\quad N=3,4, \end{align*} where \begin{align*} &L_{3}:=\{q\le 4673, ~q\ \textrm{prime}\} \cup \{5003,6007,7001,8009\},\\ &L_{4}:=\{q\le 1361, ~q\ \textrm{prime}\} \cup \{1409\}. \end{align*} Our investigations and results allow to conjecture that these bounds hold for all $q$.