Tables, bounds and graphics of sizes of complete arcs in the plane $\mathrm{PG}(2,q)$ for all $q\le321007$ and sporadic $q$ in $[323761\ldots430007]$ obtained by an algorithm with fixed order of points (FOP)

Abstract

In the previous works of the authors, a step-by-step algorithm FOP which uses any fixed order of points in the projective plane $\mathrm{PG}(2,q)$ is proposed to construct small complete arcs. In each step, the algorithm adds to a current arc the first point in the fixed order not lying on the bisecants of the arc. The algorithm is based on the intuitive postulate that $\mathrm{PG}(2,q)$ contains a sufficient number of relatively small complete arcs. Also, in the previous papers, it is shown that the type of order on the points of $\mathrm{PG}(2,q)$ is not relevant. A complete lexiarc in $\mathrm{PG}(2,q)$ is a complete arc obtained by the algorithm FOP using the lexicographical order of points. In this work, we collect and analyze the sizes of complete lexiarcs in the following regions: \begin{align*}& \textbf{all } q\le321007,~ q \mbox{ prime power}; & 15 \mbox{ sporadic $q$'s in the interval }[323761\ldots430007], \mbox{ see (1.10)}. \end{align*} In the work [9], the smallest known sizes of complete arcs in $\mathrm{PG}(2,q)$ are collected for all $q\leq160001$, $q$ prime power. The sizes of complete arcs, collected in this work and in [9], provide the following upper bounds on the smallest size $t_{2}(2,q)$ of a complete arc in the projective plane $\mathrm{PG}(2,q)$: \begin{align*} t_{2}(2,q)&<0.998\sqrt{3q\ln q}<1.729\sqrt{q\ln q}&\mbox{ for }&&7&\le q\le160001;\\ t_{2}(2,q)&<1.05\sqrt{3q\ln q}<1.819\sqrt{q\ln q}&\mbox{ for }&&7&\le q\le321007. \end{align*} Our investigations and results allow to conjecture that the bound $t_{2}(2,q)<1.05\sqrt{3q\ln q}<1.819\sqrt{q\ln q}$ holds for all $q\ge7$. It is noted that sizes of the random complete arcs and complete lexiarcs behave similarly. This work can be considered as a continuation and development of the paper [11].