Tables, bounds and graphics of sizes of complete lexiarcs in the plane $\mathrm{PG}(2,q)$ for all $q\le301813$ and sporadic $q$ in the interval $[301897\ldots430007]$ obtained by an algorithm with fixed order of points (FOP)

Daniele Bartoli, Alexander A. Davydov, Giorgio Faina, Alexey A. Kreshchuk, Stefano Marcugini, Fernanda Pambianco

Published 2014-04-02, updated 2015-07-31Version 2

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 for constructing small complete arcs. In every step, the algorithm adds to a running arc the first point in the fixed order not lying on the bisecants of the arc. In this work we $\textbf{collect the sizes of complete lexiarcs}$ (i.e. complete arcs in $\mathrm{PG}(2,q)$ obtained by the algorithm FOP with the lexicographical orders of points) in the following regions: $\textbf{all}$ $q\le301813$, $q$ prime power; 23 sporadic $q$'s in the interval $[301897\ldots430007]$, see Table 2. 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 the work [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\le301813;\\ t_{2}(2,q)&<\sqrt{q}\ln^{0.7295}q&\mbox{ for }&&109&\le q\le160001;\\ t_{2}(2,q)&<\sqrt{q}\ln^{0.7404}q&\mbox{ for }&&160001&<q\le301813. \end{align*} Our investigations and results allow to conjecture that the 2-nd and 3-rd bounds above hold for all $q\geq109$. In addition, it is shown that sizes of complete lexiarcs in $\mathrm{PG}(2,q)$, $q\gtrapprox90000$, exceed the smallest known sizes of complete arcs by only at most $6\%$. It is noted also 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 [10].