On Ovoids in Orthogonal Spaces of Type $O_5(q)$

Abstract

It is known that the ovoids in $O_5(q)$, $q\le 7$, are classical ovoids. Using algebraic and computational techniques, we classify ovoids in $O_5(9)$ and $O_5(11)$ with the aid of a computer. We also study the ovoids which contain an irreducible conic and classify them in $O_5(13)$. Our results show that there is only one nonclassical ovoid (a member from a family of Kantor) up to isomorphism in $O_5(9)$ and all the ovoids in $O_5(11)$ are classical.