## Innovations in Incidence Geometry

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

#### Abstract

We classify spreads of the Tits quadrangles $T 2 ( O )$, for $O$ an oval in $PG ( 2 , q )$, for $q = 2 , 4 , 8 , 1 6$ and $3 2$, using a computer for the last three cases. Along the way, we classify $α$-flocks of $PG ( 3 , 3 2 )$, and so flocks of the quadratic cone in $PG ( 3 , 3 2 )$. Perhaps our most striking results are that, for many ovals $O$ in $PG ( 2 , 3 2 )$, including all 12 O’Keefe-Penttila ovals, $T 2 ( O )$ has no spreads, and that $T 2 ( O )$ is a proper subGQ of a GQ of order $( s , 3 2 )$ for precisely 6 of the 35 ovals $O$ of $PG ( 2 , 3 2 )$, all of which were previously known to be subquadrangles of a (flock or dual Tits) GQ of order $( 1 0 2 4 , 3 2 )$. Also $T 2 ( O )$ 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 $q = 1 6 , 3 2$.

#### Article information

Source
Innov. Incidence Geom., Volume 6-7, Number 1 (2007), 111-126.

Dates
Accepted: 4 March 2008
First available in Project Euclid: 28 February 2019

https://projecteuclid.org/euclid.iig/1551323173

Digital Object Identifier
doi:10.2140/iig.2008.6.111

Mathematical Reviews number (MathSciNet)
MR2515261

Zentralblatt MATH identifier
1175.51003

#### Citation

Brown, Matthew R.; O’Keefe, Christine M.; Payne, Stanley E.; Penttila, Tim; Royle, Gordon F. The classification of spreads of $T_2({\mathcal O})$ and $\alpha$-flocks over small fields. Innov. Incidence Geom. 6-7 (2007), no. 1, 111--126. doi:10.2140/iig.2008.6.111. https://projecteuclid.org/euclid.iig/1551323173

#### References

• S. Ball, P. Govaerts and L. Storme, On ovoids of parabolic quadrics, Des. Codes Cryptogr. 38 (2006), 131–145.
• R. C. Bose, Mathematical theory of the symmetrical factorial design, Sankhyā 8 (1947), 107–166.
• W. Bosma, J. Cannon, and C. Playoust, The Magma algebra system. I. The user language, J. Symbolic Comput. 24 (1997), 235–265.
• M. R. Brown, The determination of ovoids of $\pg(3,q)$ containing a pointed conic. Second Pythagorean Conference (Pythagoreion, 1999), J. Geom. 67 (2000), 61–72.
• M. R. Brown and M. Lavrauw, Eggs in $\pg(4n-1,q)$, $q$ even, containing a pseudo-pointed conic, European J. Combin. 26 (2005), no. 1, 117–128.
• M. R. Brown, C. M. O'Keefe, S. E. Payne, T. Penttila and G. F. Royle, Spreads of $T_2(\O)$, $\alpha$-flocks and ovals, Des. Codes Cryptogr. 31 (2004), 251–282.
• M. R. Brown, I. Pinneri and G. F. Royle, private communication, 1996.
• I. Cardinali and S. E. Payne, $q$-Clan Geometries in Characteristic 2, Frontiers in Mathematics, Birkhäuser Verlag, Basel, 2007.
• W. Cherowitzo, $\alpha$-Flocks and hyperovals, Geom. Dedicata 72 (1998), 221–246.
• F. De Clerck and C. Herssens, Flocks of the quadratic cone in $\pg(3,q)$, for $q$ small, The CAGe reports 8, Computer Algebra Group, The University of Gent, Ghent, Belgium.
• G. Fellegara, Gli ovaloidi in uno spazio tridimensionale di Galois di ordine $8$, Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Nat. (8) 32 (1962), 170–176.
• J. C. Fisher and J. A. Thas, Flocks in $\pg(3,q)$, Math. Z. 169 (1979), 1–11.
• J. Fuelberth and A. Gunawardena, On ovoids in orthogonal spaces of type ${\rm O}\sb 5(q)$, J. Combin. Math. Combin. Comput. 29 (1999), 79–86.
• M. Hall, Jr., Ovals in the Desarguesian plane of order $16$, Ann. Mat. Pura Appl. (4) 102 (1975), 159–176.
• C. M. O'Keefe and T. Penttila, Ovoids of $\pg(3,16)$ are elliptic quadrics, J. Geom. 44 (1990), 95–106.
• ––––, Hyperovals in $\pg(2,16)$. European J. Combin. 12 (1991), 51–59.
• ––––, Ovoids of $\pg(3,16)$ are elliptic quadrics, II, J. Geom. 44 (1992), no. 1-2, 140–159.
• ––––, Automorphism groups of generalized quadrangles via an unusual action of $\mathsf{P\Gamma L}(2,2\sp h)$. European J. Combin. 23 (2002), 213–232.
• C. M. O'Keefe, T. Penttila and G. F. Royle, Classification of ovoids in $\pg(3,32)$, J. Geom 50 no. 1–2 (1994), 143–150.
• S. E. Payne and J. A. Thas, Finite Generalized Quadrangles, Pitman, London, 1984.
• T. Penttila and C. E. Praeger, Ovoids and translation ovals, J. London Math. Soc. $(2)$ 56, no. 3 (1997), 607–624.
• T. Penttila and G.F. Royle, Classification of hyperovals in $\pg(2,32)$, J. Geom. 50 (1994), 151–158.
• T. Penttila and B. Williams, Ovoids of parabolic spaces, Geom. Dedicata 82 (2000), 1–19.
• I. Pinneri, Flocks, Generalised Quadrangles and Hyperovals, Ph.D. Thesis, University of Western Australia, 1996.
• B. Segre, Sui $k$-archi nei piani finiti di caratteristica due, Rev. Math. Pures Appl. 2 (1957), 289–300.
• E. Seiden, A theorem in finite projective geometry and an application to statistics, Proc. Amer. Math. Soc. 1, (1950), 282–286.
• J. A. Thas, Generalized quadrangles and flocks of cones, European J. Combin. 8 (1987), no. 4, 441–452.
• J. A. Thas and S. E. Payne, Spreads and ovoids in finite generalized quadrangles, Geom. Dedicata 52 (1994), 227–253.
• J. Tits, Ovoides et groupes du Suzuki, Arch. Math. 13 (1962), 187–198.
• B. Williams, Ovoids of parabolic and hyperbolic spaces, Ph.D. Thesis, University of Western Australia, 1998.