Tables, bounds and graphics of the smallest known sizes of complete arcs in the plane $\mathrm{P}\mathrm{G}(2,q)$ for all $q\le 160001$ and sporadic $q$ in the interval $[160801\dots 430007]$

In the projective planes $\mathrm{P}\mathrm{G}(2,q)$, we collect the smallest known sizes of complete arcs for the regions \begin{align*} &\mbox{all } q\le160001,~~ q \mbox{ prime power};\ &Q_{4}={34 \mbox{ sporadic }q’\mbox{s in the interval }[160801\ldots430007], \mbox{ see Table 3}}. \end{align*} For $q\le 160001$, the collection of arc sizes is complete in the sense that arcs for all prime powers are considered. This proves new upper bounds on the smallest size $t_2(2,q)$ of a complete arc in $\mathrm{P}\mathrm{G}(2,q)$, in particular \begin{align*} t_{2}(2,q)&<0.998\sqrt{3q\ln q}<1.729\sqrt{q\ln q}&\mbox{ for }&&7&\le q\le160001;(1) \ t_{2}(2,q)&<\sqrt{q}\ln^{0.7295}q&\mbox{ for }&&109&\le q\le160001;(2)\ t_{2}(2,q)&<\sqrt{q}\ln^{c_{up}(q)}q,c_{up}(q)=\frac{0.27}{\ln q}+0.7,&\mbox{ for }&&19&\le q\le160001;(3)\ t_{2}(2,q)&<0.6\sqrt{q}\ln^{\varphi_{up}(q;0.6)} q,\varphi_{up}(q;0.6)=\frac{1.5}{\ln q}+0.802,&\mbox{ for }&&19&\le q\le160001.(4) \end{align*} Moreover, the bounds (2) – (4) hold also for $q\in Q_4$. Also, \begin{align*} t_{2}(2,q)&<1.006\sqrt{3q\ln q}<1.743\sqrt{q\ln q}&\mbox{ for }&&q\in Q_{4}.~~(5) \end{align*} Our investigations and results allow to conjecture that the bounds (2) – (5) hold for all $q\ge 109$.