On dimensional dual hyperovals ${\mathcal S}^{d+1}_{\sigma,\phi}$

Abstract

A $d$-dimensional dual hyperoval ${\mathcal{S}}_{\sigma ,\varphi }^{d+1}$ inside $PG\left(2d+1,2\right)$ ($d\ge 2$) was constructed by Yoshiara, for a generator $\sigma$ of Gal$\left(GF\left(q\right)∕GF\left(2\right)\right)$ and an o-polynomial $\varphi \left(X\right)$ of $GF\left(q\right)\left[X\right]$ ($q=2^{d+1}$). There, its automorphism group is determined and a criterion is given for these dimensional dual hyperovals to be isomorphic, assuming that the map $\varphi$ on $GF\left(q\right)$ induced by $\varphi \left(X\right)$ lies in Gal$\left(GF\left(q\right)∕GF\left(2\right)\right)$. In this paper, we extend these results for a monomial o-polynomial $\varphi$. We show that Aut$\left({\mathcal{S}}_{\sigma ,\varphi }^{d+1}\right)\cong G{L}_3\left(2\right)$ or $Z_{q-1}.Z_{d+1}$ according as $d=2$ or $d\ge 3$, if $\varphi \left(X\right)$ is monomial but $\varphi \notin$Gal$\left(GF\left(q\right)∕GF\left(2\right)\right)$. In particular, a special member $X\left(0\right)$ of ${\mathcal{S}}_{\sigma ,\varphi }^{d+1}$ is always fixed by any automorphism of ${\mathcal{S}}_{\sigma ,\varphi }^{d+1}$. Furthermore, ${\mathcal{S}}_{\sigma ,\varphi }^{d+1}\cong {\mathcal{S}}_{{\sigma }^{\prime },{\varphi }^{\prime }}^{d+1}$ if and only if either $\left(\sigma ,\varphi \right)=\left({\sigma }^{\prime },{\varphi }^{\prime }\right)$ or $\sigma {\sigma }^{\prime }=\varphi {\varphi }^{\prime }=$id.