New quotients of the $d$-dimensional Veronesean dual hyperoval in $\mathrm{PG}(2d+1,2)$

Abstract

Let $d\ge 3$. For each $e\ge 1$, Thas and Van Maldeghem constructed a $d$-dimensional dual hyperoval in $PG\left(d\left(d+3\right)∕2,q\right)$ with $q=2^e$, called the Veronesean dual hyperoval. A quotient of the Veronesean dual hyperoval with ambient space $PG\left(2d+1,q\right)$, denoted $S_{\sigma }$, is constructed by Taniguchi, using a generator $\sigma$ of the Galois group Gal$\left(GF\left(q^{d+1}\right)∕GF\left(q\right)\right)$.

In this note, using the above generator $\sigma$ for $q=2$ and a $d$-dimensional vector subspace $H$ of $GF\left(2^{d+1}\right)$ over $GF\left(2\right)$, we construct a quotient $S_{\sigma ,H}$ of the Veronesean dual hyperoval in $PG\left(2d+1,2\right)$ in case $d$ is even. Moreover, we prove the following: for generators $\sigma$ and $\tau$ of the Galois group Gal$\left(GF\left(2^{d+1}\right)∕GF\left(2\right)\right)$,

1. $S_{\sigma }$ above (for $q=2$) is not isomorphic to $S_{\tau ,H}$,

2. $S_{\sigma ,H}$ is isomorphic to $S_{\sigma ,H^{{}^{\prime }}}$ for any $d$-dimensional vector subspaces $H$ and $H^{{}^{\prime }}$ of $GF\left(2^{d+1}\right)$, and

3. $S_{\sigma ,H}$ is isomorphic to $S_{\tau ,H}$ if and only if $\sigma =\tau$ or $\sigma ={\tau }^{-1}$.

Hence, we construct many new non-isomorphic quotients of the Veronesean dual hyperoval in $PG\left(2d+1,2\right)$.