Abstract

We classify spreads of the Tits quadrangles $T_2\left(\mathcal{O}\right)$, for $\mathcal{O}$ an oval in $\mathsf{PG}\left(2,q\right)$, for $q=2,4,8,16$ and $32$, using a computer for the last three cases. Along the way, we classify $\alpha$-flocks of $\mathsf{PG}\left(3,32\right)$, and so flocks of the quadratic cone in $\mathsf{PG}\left(3,32\right)$. Perhaps our most striking results are that, for many ovals $\mathcal{O}$ in $\mathsf{PG}\left(2,32\right)$, including all 12 O’Keefe-Penttila ovals, $T_2\left(\mathcal{O}\right)$ has no spreads, and that $T_2\left(\mathcal{O}\right)$ is a proper subGQ of a GQ of order $\left(s,32\right)$ for precisely 6 of the 35 ovals $\mathcal{O}$ of $\mathsf{PG}\left(2,32\right)$, all of which were previously known to be subquadrangles of a (flock or dual Tits) GQ of order $\left(1024,32\right)$. Also $T_2\left(\mathcal{O}\right)$ is not a proper subGQ of a GQ of order $\left(s,q\right)$ or of a GQ of order $\left(q,t\right)$ for $\mathcal{O}$ a pointed conic in $\mathsf{PG}\left(2,q\right)$, for $q=16,32$.

Keywords

Mathematical Subject Classification 2000

Milestones

Authors