Let
$d\ge 3$. For each
$e\ge 1$, Thas and Van Maldeghem
constructed a
$d$dimensional
dual hyperoval in
$PG\left(d\left(d+3\right)\u22152,q\right)$
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$\left(GF\left({q}^{d+1}\right)\u2215GF\left(q\right)\right)$.
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$\left(GF\left({2}^{d+1}\right)\u2215GF\left(2\right)\right)$,
 ${S}_{\sigma}$
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 nonisomorphic quotients of the Veronesean dual hyperoval
in
$PG\left(2d+1,2\right)$.
