Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: 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)   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].

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The categories of this paper are: funct-an
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The categories of this paper are: comp-gas
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The categories of this paper are: cond-mat
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The categories of this paper are: hep-lat
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The categories of this paper are: acc-phys
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The categories of this paper are: hep-ex
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The categories of this paper are: astro-ph
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The categories of this paper are: nucl-th
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The categories of this paper are: hep-ph
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The categories of this paper are: q-fin
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The categories of this paper are: cs
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The categories of this paper are: hep-th
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The categories of this paper are: atom-ph
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The categories of this paper are: econ
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The categories of this paper are: gr-qc
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The categories of this paper are: nucl-ex