A new approach to arcs (Q1964612)

The authors consider the Desarguesian plane \(\text{PG}(2,q)\) represented via a difference set in the cyclic group of order \(q^2+q+1\), and define the covering number of a set \(S\) of points as the smallest number of lines needed to cover the set \(-S\). They use this notion to study \(k\)-arcs and call such an arc flat when its covering number is 1. In particular, they focus on hyperovals in even order planes and consider those with covering numbers 1 and 2. Small order planes are examined and some classical results are recovered. An algorithm is suggested which tests whether a \(k\)-set of points forms an arc.