On Translation Hyperovals in Semifield Planes

TL;DR

This paper presents the first known example of a finite translation plane lacking a translation hyperoval, challenging previous conjectures, and links this to properties of related rank-metric codes and scattered subspaces.

Contribution

It provides the first counterexample of a translation plane without a translation hyperoval, specifically a semifield plane of order 64, and connects this to coding theory and geometric structures.

Findings

Counterexample of a translation plane without a translation hyperoval

Relation between hyperoval non-existence and rank-metric code properties

Implication for the structure of semifield planes

Abstract

In this paper we demonstrate the first example of a finite translation plane which does not contain a translation hyperoval, disproving a conjecture of Cherowitzo. The counterexample is a semifield plane, specifically a Generalised Twisted Field plane, of order $64$. We also relate this non-existence to the covering radius of two associated rank-metric codes, and the non-existence of scattered subspaces of maximum dimension with respect to the associated spread.