Maximal arcs and extended cyclic codes

TL;DR

The paper proves the existence of certain Denniston maximal arcs in projective planes over finite fields, invariant under cyclic groups, using geometric and coding-theoretic methods.

Contribution

It establishes the existence of cyclically invariant Denniston maximal arcs for specific degrees, providing two different proofs.

Findings

Existence of Denniston maximal arcs for degrees dividing q-1.

Construction of arcs invariant under cyclic groups fixing one point.

Connection between maximal arcs and two-weight 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.