Translation Hyperovals and $\mathbb{F}_2$-Linear Sets of Pseudoregulus Type

Abstract

In this paper, we study translation hyperovals in PG(2,q^k). The main result of this paper characterises the point sets defined by translation hyperovals in the André/Bruck-Bose representation. We show that the affine point sets of translation hyperovals in PG(2,q^k) are precisely those that have a scattered F₂-linear set of pseudoregulus type in PG(2k−1,q) as set of directions. This correspondence is used to generalise the results of Barwick and Jackson who provided a characterisation for translation hyperovals in PG(2,q²), see [S.G. Barwick, Wen-Ai Jackson, A characterization of translation ovals in finite even order planes. Finite fields Appl. 33 (2015), 37--52.].