The classification of maximal arcs in small Desarguesian planes

Abstract

There are three types of maximal arcs in the planes of order 16, the hy- perovals of degree 2, the dual hyperovals of degree 8 and the maximal arcs of degree 4. The hyperovals and dual hyperovals of the Desarguesian projective plane PG(2; q) have been classi??ed for q ?? 32. This article completes the classi??cation of maximal arcs in PG(2; 16). The initial calculations are valid for all maximal arcs of degree r in PG(2; q). In the case r = q=4 (dually r = 4) further computations are possible. By means of a precursor we classify the hyperovals in PG(2; 8) using these calculations and then classify, with the aid of a computer, the maximal arcs of degree 4 in PG(2; 16); they are all Denniston maximal arcs.