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

We classify spreads of the Tits quadrangles T2(O), for O an oval in PG(2,q), for q = 2,4,8,16 and 32, using a computer for the last three cases. Along the way, we classify α-flocks of PG(3,32), and so flocks of the quadratic cone in PG(3,32). Perhaps our most striking results are that, for many ovals O in PG(2,32), including all 12 O’Keefe-Penttila ovals, T2(O) has no spreads, and that T2(O) is a proper subGQ of a GQ of order (s,32) for precisely 6 of the 35 ovals O of PG(2,32), all of which were previously known to be subquadrangles of a (flock or dual Tits) GQ of order (1024,32). Also T2(O) is not a proper subGQ of a GQ of order (s,q) or of a GQ of order (q,t) for O a pointed conic in PG(2,q), 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