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

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 $q$.