更新：关于$(n,r)$下界的一些新结果-$q\le 31$的$PG(2,q)$中的弧
Update: Some new results on lower bounds on $(n,r)$-arcs in $PG(2,q)$ for $q\le 31$
摘要：
$PG(2,q)$中的$(n,r)$ -弧是$PG(2,q)$中的点的集合$B$，使得$PG(2,q)$中的每条线包含至多$B$的$r$元素，并且使得存在至少一条线恰好包含$B$的$r$元素。值$m_r(2,q)$表示存在$(n,r)$ -弧的射影几何$PG(2,q)$中的点的最大数目$n$。通过使用指定的自同构和整数线性规划显式构造$(n,r)$ -弧，我们得到了$m_r(2,q)$的一些改进的下界：$m_{10}(2,16)\ge 144$，$m_3(2,25)\ge 39$，$m_{18}(2,25)\ge 418$，$m_9(2,27)\ge 201$，$m_{14}(2,29)\ge 364$，$m_{25}(2,29)\ge 697$，$m_{25}(2,31)\ge 734$ .此外，我们表明，通过系统地排除可能的自同构，假定$(44,5)$ -弧，$(90,9)$ -弧在$PG(2,11)$，$(39,4)$ -弧在$PG(2,13)$ -在它们存在的情况下-是刚性的，即它们都只承认平凡的自同构群的顺序$1$。此外，$PG(2,13)$中假定的$(50,5)$ -弧、$(65,6)$ -弧、$(119,10)$ -弧、$(133,11)$ -弧和$(146,12)$ -弧将是刚性的，或者将允许唯一的$2$阶自同构群（直到共轭）。
英文摘要：
An $(n,r)$-arc in $PG(2,q)$ is a set $B$ of points in $PG(2,q)$ such that each line in $PG(2,q)$ contains at most $r$ elements of $B$ and such that there is at least one line containing exactly $r$ elements of $B$. The value $m_r(2,q)$ denotes the maximal number $n$ of points in the projective geometry $PG(2,q)$ for which an $(n,r)$-arc exists. By explicitly constructing $(n,r)$-arcs using prescribed automorphisms and integer linear programming we obtain some improved lower bounds for $m_r(2,q)$: $m_{10}(2,16)\ge 144$, $m_3(2,25)\ge 39$, $m_{18}(2,25)\ge 418$, $m_9(2,27)\ge 201$, $m_{14}(2,29)\ge 364$, $m_{25}(2,29)\ge 697$, $m_{25}(2,31)\ge 734$. Furthermore, we show by systematically excluding possible automorphisms that putative $(44,5)$-arcs, $(90,9)$-arcs in $PG(2,11)$, and $(39,4)$-arcs in $PG(2,13)$ -- in case of their existence -- are rigid, i.e. they all would only admit the trivial automorphism group of order $1$. In addition, putative $(50,5)$-arcs, $(65,6)$-arcs, $(119,10)$-arcs, $(133,11)$-arcs, and $(146,12)$-arcs in $PG(2,13)$ would be rigid or would admit a unique automorphism group (up to conjugation) of order $2$.
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