#### Vol. 12, No. 1, 2011

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

### Hiroaki Taniguchi and Satoshi Yoshiara

Vol. 12 (2011), No. 1, 151–165
##### 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$\left(GF\left({2}^{d+1}\right)∕GF\left(2\right)\right)$,

• ${S}_{\sigma }$ above (for $q=2$) is not isomorphic to ${S}_{\tau ,H}$,
• ${S}_{\sigma ,H}$ 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
• ${S}_{\sigma ,H}$ 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 $PG\left(2d+1,2\right)$.

##### Keywords
dual hyperoval, Veronesean, quotient
##### Mathematical Subject Classification 2010
Primary: 05BXX, 05EXX, 51EXX