  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 $2$.
