Bounds on partial ovoids and spreads in classical generalized quadrangles

Abstract

We present an improvement on a recent bound for small maximal partial ovoids of $\mathbf{W}\left(q^3\right)$. We also classify maximal partial ovoids of size $\left(q^2-1\right)$ of $\mathbf{Q}\left(4,q\right)$ which allow a certain large automorphism group, and discuss examples for small $q$.