= Suzuki ovoid
{c}
{title2=$|\mathcal O|=q^2+1$}
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 $q\geq8$, 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 $\binom{q^2+1}{2}$ such secants. This turns part of the geometry into an explicit pair-counting problem.<On the simple groups of Suzuki and Ree by Robert A. Wilson>[Introduction and section4.1, Suzuki ovoid and natural doubly transitive action]{ref}
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