Hyperovals in Knuth's binary semifield planes

Abstract

In each of the three projective planes coordinatised by the Knuth's binary semifield ${\mathbb{K}}_n$ of order $2^n$ and two of its Knuth derivatives, we exhibit a new family of infinitely many translation hyperovals. In particular, when $n=5$, we also present complete lists of all translation hyperovals in them. The properties of some designs associated with these hyperovals are also studied.