Small maximal partial ovoids of $H(3,q^2)$

Abstract

The trivial lower bound for the size of a maximal partial ovoid of $H\left(3,q^2\right)$ is $q^2+1$. Ebert showed that this bound can be attained if and only if $q$ is even. In the present paper it is shown that a maximal partial ovoid of $H\left(3,q^2\right)$, $q$ odd, has at least $q^2+1+\frac{4}{9}q$ points (previously, only $q^2+3$ was known). It is also shown that a maximal partial spread of $H\left(3,q^2\right)$, $q$ even, has size $q^2+1$ or size at least $q^2+1+\frac{4}{9}q$.