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In the previous works of the authors, a step-by-step algorithm FOP which usesany fixed order of points in the projective plane $\mathrm{PG}(2,q)$ isproposed to construct small complete arcs. In each step, the algorithm adds toa current arc the first point in the fixed order not lying on the bisecants ofthe arc. The algorithm is based on the intuitive postulate that$\mathrm{PG}(2,q)$ contains a sufficient number of relatively small completearcs. Also, in the previous papers, it is shown that the type of order on thepoints 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 thelexicographical order of points. In this work, we collect and analyze the sizesof 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], thesmallest known sizes of complete arcs in $\mathrm{PG}(2,q)$ are collected forall $q\leq160001$, $q$ prime power. The sizes of complete arcs, collected inthis 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 toconjecture 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 andcomplete lexiarcs behave similarly. This work can be considered as acontinuation and development of the paper [11].

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)