Elation KM-arcs

Abstract

In this paper, we study KM-arcs in $PG(2, q)$, the Desarguesian projective plane of order $q$. A KM-arc A of type $t$ is a natural generalisation of a hyperoval: it is a set of $q + t$ points in $PG(2, q)$ such that every line of $PG(2,q)$ meets A in $0,2$ or $t$ points. We study a particular class of KM-arcs, namely, elation KM-arcs. These KM-arcs are highly symmetrical and moreover, many of the known examples are elation KM-arcs. We provide an algebraic framework and show that all elation KM-arcs of type $q/4$ in $PG(2,q)$ are translation KM-arcs. Using a result of [2], this concludes the classification problem for elation KM-arcs of type $q/4$. Furthermore, we construct for all $q = 2^h$, $h > 3$, an infinite family of elation KM-arcs of type $q/8$, and for $q = 2^h$, where $4, 6, 7 | h$ an infinite family of KM-arcs of type $q/16$. Both families contain new examples of KM-arcs.