Update: Some new results on lower bounds on $(n,r)$-arcs in $PG(2,q)$ for $q\le 31$

Abstract

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 $2$.