Hyperovals in PG(2,4) and a self-dual code (Q1808810)

In \(PG(2,4)\), a 2-intersecting family \(S\) is a set of hyperovals that pairwise intersect in exactly two points. This paper contains two separate results. The first result gives the size and structure of a set of hyperovals \(S\) of maximum cardinality in \(PG(2,4)\). In particular, the following theorem is proved: if \(S\) is a 2-intersecting family of maximum cardinality, then \(|S|=16\). Moreover, either all the hyperovals in \(S\) have a common point, or else all the hyperovals in \(S\) are skew to some line of \(PG(2,4)\). The second result exhibits a self dual \([4^{2s},4^{2s}/2]\)-code which can be obtained from the geometry of \(PG(2,4^s)\), for each \(s\geq 1\).