Suzuki ovoid ($|\mathcal{O}|=q^2+1$)

A Suzuki ovoid is a special set of $q^2+1$ points in the three-dimensional projective space over ${\mathbb{F}}_q$, for $q=2^{2a+1}$. It is an ovoid: no three of its points are collinear. Its projective symmetries include the Suzuki group of Lie type $Sz(q)$, acting two-transitively on these points.

For $q\ge 8$, fixing one point leaves a stabilizer of order $q^2(q-1)$, with a subgroup of order $q^2$ acting regularly on the other $q^2$ points. Thus the geometry realizes the natural permutation action of the group, rather than only its four-dimensional matrix action. For $q=8$, there are $65$ ovoid points and $|Sz(8)|=29120$.

The absence of collinear triples means that choosing two ovoid points determines a secant containing exactly those two ovoid points. There are therefore $\left(\genfrac{}{}{0ex}{}{q^2+1}{2}\right)$ such secants. This turns part of the geometry into an explicit pair-counting problem.^[ref]