The classification of maximal arcs in small Desarguesian planes (Q1420721)

A \((k, r)\)-arc in a projective plane is a non-empty set of \(k\) points, at most \(r\) on every line. If the order of the plane is \(q\), then \(k\leq 1+ (q +1)(r- 1)\). Arcs realizing the upper bound are called maximal arcs and \(r\) is called the degree of the maximal arc. From authors' abstract: ``There are three types of maximal arcs in the plane of order 16, the hyperovals 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 \(\text{PG}(2, q)\) have been classified for \(q\leq 32\). This article completes the classification of maximal arcs in \(\text{PG}(2, 16)\). The initial calculations are valid for all maximal arcs of degree \(r\) in \(\text{PG}(2, q)\). In the case \(r= q/4\) (dually \(r= 4\)) further computations are possible. By means of a precursor we classify the hyperovals in \(\text{PG}(2, 8)\) using these calculations and then classify, with the aid of a computer, the maximal arcs of degree 4 in \(\text{PG}(2,16)\); they are all Denniston maximal arcs.''