Dimensional dual hyperovals associated with quadratic APN functions

Abstract

From each quadratic APN function $f$ on $GF\left(2^{d+1}\right)$ with $f\left(0\right)=0$, we construct a $d$-dimensional dual hyperoval ${\mathcal{S}}^{d+1}\left[f\right]$ over $GF\left(2\right)$. This provides many new examples of $d$-dual hyperovals over $GF\left(2\right)$ with ambient spaces of dimension $2d+2$. The automorphism group of ${\mathcal{S}}^{d+1}\left[f\right]$ is a semidirect product of a normal subgroup acting regularly on the members of ${\mathcal{S}}^{d+1}\left[f\right]$ with the stabilizer of a member. Some methods are provided to analyse the structure of the stabilizer. Based on them, $Aut\left({\mathcal{S}}^{10}\left[f\right]\right)$ for an APN function $f$ on $GF\left(2^{10}\right)$ found in by Edel, Kyureghyan and Pott is determined.