Hyperovals in Knuth's binary semifield planes

TL;DR

This paper introduces a new infinite family of translation hyperovals in projective planes derived from Knuth's binary semifield, providing complete classifications for specific cases and exploring related design properties.

Contribution

It presents the first construction of infinitely many translation hyperovals in these planes and offers complete classifications for the case when n=5.

Findings

Existence of infinitely many translation hyperovals in Knuth's semifield planes.

Complete classification of hyperovals for n=5.

Analysis of associated combinatorial designs.

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.