Construction of $(n,r)$-arcs in $\mathrm{PG}(2,q)$

Abstract

We construct new $\left(n,r\right)$-arcs in PG$\left(2,q\right)$ by prescribing a group of automorphisms and solving the resulting Diophantine linear system with lattice point enumeration. We can improve the known lower bounds for $q=11,13,16,17,19$ and give the first example of a double blocking set of size $n$ in PG$\left(2,p\right)$ with $n<3p$ and $p$ prime.