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

TL;DR

This paper improves lower bounds on the maximum size of certain point sets in projective planes over finite fields using explicit constructions and automorphism analysis.

Contribution

It provides new lower bounds for $m_r(2,q)$ in $PG(2,q)$ for various q and r, and analyzes automorphism groups of hypothetical arcs.

Findings

Improved lower bounds for $m_r(2,q)$ in several cases.

Most considered arcs are shown to be rigid or have limited automorphism groups.

Explicit constructions using automorphisms and linear programming.

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 $2$.