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 \( q = 11 \).