对于区间$[160801\ldots 430007]$中的所有$q\le160001$和零星$q$，平面$\mathrm{PG}(2,q)$中完整弧的最小已知尺寸的表、界限和图形
Tables, bounds and graphics of the smallest known sizes of complete arcs in the plane $\mathrm{PG}(2,q)$ for all $q\le160001$ and sporadic $q$ in the interval $[160801\ldots 430007]$
摘要：
在投影平面$\mathrm{PG}(2,q)$中，我们收集区域\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*}的完整弧的最小已知尺寸。对于$q\le160001$，弧尺寸的收集在考虑所有素数幂的弧的意义上是完整的。这证明了$\mathrm{PG}(2,q)$，特别是\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^{φ_{up}(q;0.6)} q,~~φ_{up}(q;0.6)=\frac{1.5}{\ln q}+0.802,&\mbox{ for }&&19&\le q\le160001.~~(4) \end{align*}中完整弧的最小尺寸$t_{2}(2,q)$的新的上界。此外，界（2）-（4）对$q\in Q_{4}$也成立。此外，\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*}我们的研究和结果允许推测界（2）-（5）对所有$q\geq109$成立。
英文摘要：
In the projective planes $\mathrm{PG}(2,q)$, 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 $q\le160001$, 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 $\mathrm{PG}(2,q)$, 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^{φ_{up}(q;0.6)} q,~~φ_{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 $q\geq109$.
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