#### Vol. 6+7, No. 1, 2008

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The classification of spreads of $T_2({\mathcal O})$ and $\alpha$-flocks over small fields

### Matthew R. Brown, Christine Margaret O’Keefe, Stanley E. Payne, Tim Penttila and Gordon F. Royle

Vol. 6+7 (2008), No. 1, 111–126
##### Abstract

We classify spreads of the Tits quadrangles ${T}_{2}\left(\mathsc{O}\right)$, for $\mathsc{O}$ 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 $\mathsc{O}$ in $\mathsf{PG}\left(2,32\right)$, including all 12 O’Keefe-Penttila ovals, ${T}_{2}\left(\mathsc{O}\right)$ has no spreads, and that ${T}_{2}\left(\mathsc{O}\right)$ is a proper subGQ of a GQ of order $\left(s,32\right)$ for precisely 6 of the 35 ovals $\mathsc{O}$ 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 ${T}_{2}\left(\mathsc{O}\right)$ 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 $\mathsc{O}$ a pointed conic in $\mathsf{PG}\left(2,q\right)$, for $q=16,32$.

##### Keywords
generalized quadrangle, spread, flock, subquadrangle, oval
##### Mathematical Subject Classification 2000
Primary: 51B15, 51E12, 51E20, 51E21, 51E23
##### Milestones
Received: 27 December 2007
Accepted: 4 March 2008
##### Authors
 Matthew R. Brown Christine Margaret O’Keefe Stanley E. Payne Tim Penttila Gordon F. Royle