Large Arcs in Small Plan

Abstract

In a finite projective plane $\text{PG}(2,q)$, a set of $k$ points is called a $(k,n)$-arc if the following two properties hold:
1. Every line intersects it in at most $n$ points.
2. There exists a line which intersects it in exactly $n$ points.

We are interested in determining, for each $q$ and each $n$, the largest value of $k$ for which a $(k,n)$-arc exists in $\text{PG}(2,q)$. If possible, we would like to classify those arcs up to isomorphism. We look at the problem for $q=11$.