Maximal arcs and extended cyclic codes

Abstract

It is proved that for every $d\ge 2$ such that $d-1$ divides $q-1$, where $q$ is a power of 2, there exists a Denniston maximal arc $A$ of degree $d$ in $\PG(2,q)$, being invariant under a cyclic linear group that fixes one point of $A$ and acts regularly on the set of the remaining points of $A$. Two alternative proofs are given, one geometric proof based on Abatangelo-Larato's characterization of Denniston arcs, and a second coding-theoretical proof based on cyclotomy and the link between maximal arcs and two-weight codes.