Let $m$ be a positive integer, $q$ be a prime power, and $\mathrm{PG}(2,q)$ be the projective plane over the finite field $\mathbb F_q$. Finding complete $m$-arcs in $\mathrm{PG}(2,q)$ of size less than $q$ is a classical problem in finite geometry. In this paper we give a complete answer to this problem when $q$ is relatively large compared with $m$, explicitly constructing the smallest $m$-arcs in the literature so far for any $m\geq 8$. For any fixed $m$, our arcs $\mathcal A_{q,m}$ satisfy $|\mathcal A_{q,m}|-q\rightarrow -\infty$ as $q$ grows. To produce such $m$-arcs, we develop a Galois theoretical machinery that allows the transfer of geometric information of points external to the arc, to arithmetic one, which in turn allows to prove the $m$-completeness of the arc.
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)
在作者以前的工作中，提出了一个逐步算法FOP，它使用任意固定顺序的点在投影平面$\mathrm{PG}(2,q)$。在每一步中，该算法将不位于弧的二等分线上的固定顺序的第一点添加到当前弧。该算法是基于直观的假设，$\mathrm{PG}(2,q)$包含足够数量的相对较小的完整的弧。此外，在以前的文件中，它表明，在$\mathrm{PG}(2,q)$的点的顺序的类型是不相关的。$\mathrm{PG}(2,q)$中的完备lexiarc是通过算法FOP使用点的字典序获得的完备弧。在这项工作中，我们收集并分析了以下区域中的完整lexiarcs的大小：\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*}在工作[9]中，收集了所有$q\leq160001$，$q$素数幂的$\mathrm{PG}(2,q)$中完整弧的最小已知大小。完整的弧的大小，收集在这项工作中，在[9]，提供了以下上界的最小尺寸$t_{2}(2,q)$的一个完整的弧在投影平面$\mathrm{PG}(2,q)$：\begin{align*} t_{2}…
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 $[160801\ldots 430007]$
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\s…
