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Spreads of T 2(o), α-flocks and Ovals

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Abstract

We establish a representation of a spread of the generalized quadrangle T 2(0), o an oval of PG(2, q), q even, in terms of a certain family of q ovals of PG(2, q) and investigate the properties of this representation. Using this representation we show that to every flock of a translation oval cone in PG(3, q) (α-flock), q even, there corresponds a spread of T 2(o) for an oval o determined by the α-flock. This gives constructions of new spreads of T 2(o), for certain ovals o, and in some cases solves the existence problem for spreads. It also provides a geometrical characterization of the ovals associated with a flock of a quadratic cone.

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References

  1. M. R. Brown, C. M. O'Keefe, S. E. Payne, T. Penttila and G. F. Royle, The classification of spreads of T 2(O) and α-flocks for q = 32; in preparation.

  2. M. R. Brown, I. Pinneri and G. F. Royle Personal communications.

  3. W. Cherowitzo, α-flocks and hyperovals, Geom. Dedicata, Vol. 72 (1998) pp. 221–246.

    Google Scholar 

  4. W. Cherowitzo, T. Penttila, I. Pinneri and G. F. Royle, Flocks and ovals, Geom. Dedicata, Vol. 60, No. 1 (1996) pp. 17–37.

    Google Scholar 

  5. P. Dembowski, Finite Geometries, Springer-Verlag, Berlin, Heidelberg, New York (1968).

    Google Scholar 

  6. D. Glynn, Plane representations of ovoids, Bull. Belg. Math. Soc. Simon Stevin, Vol. 5 (1998) pp. 275–286.

    Google Scholar 

  7. J. W. P. Hirschfeld, Projective Geometries over Finite Fields, Second Edition, Oxford University Press, Oxford (1998).

    Google Scholar 

  8. C. M. O'Keefe and T. Penttila, Ovoids of PG(3, 16) are elliptic quadrics, J. Geom., Vol. 44 (1990) pp. 95–106.

    Google Scholar 

  9. C. M. O'Keefe and T. Penttila, Ovoids of PG(3, 16) are elliptic quadrics, II, J. Geom., Vol. 44, No. 1-2 (1992) pp. 140–159.

    Google Scholar 

  10. C. M. O'Keefe and T. Penttila, A new hyperoval in PG(2, 32), J. Geom., Vol. 44 (1992) pp. 117–139.

    Google Scholar 

  11. C. M. O'Keefe and T. Penttila, Ovals with a pencil of translation ovals, Geom. Dedicata, Vol. 62 (1996) pp. 19–34.

    Google Scholar 

  12. C. M. O'Keefe and T. Penttila, Ovals in translation hyperovals and ovoids, Europ. J. Combin., Vol. 18 (1997) pp. 667–683.

    Google Scholar 

  13. C. M. O'Keefe and T. Penttila, On subquadrangles of generalized quadrangles of order (q 2; q); q even, J. Combin. Theory Ser. A, Vol. 94, No. 2 (2001) pp. 218–229.

    Google Scholar 

  14. C. M. O'Keefe, T. Penttila and G. F. Royle, Classification of ovoids in PG(3, 32), J. Geom, Vol. 50, No. 1-2 (1994) pp. 143–150.

    Google Scholar 

  15. S. E. Payne, Symmetric representations of nondegenerate generalized n-gons, Proc. Amer. Math. Soc., Vol. 19 (1968) pp. 1321–1326.

    Google Scholar 

  16. S. E. Payne, A complete determination of translation ovoids in finite Desarguesian planes, Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur. (8), Vol. 51 (1971) pp. 328–331.

    Google Scholar 

  17. S. E. Payne, Generalized quadrangles as group coset geometries, Congr. Numer., Vol. 29 (1980) pp. 717–734.

    Google Scholar 

  18. S. E. Payne, A new infinite family of generalized quadrangles, Congr. Numer., Vol. 49 (1985) pp. 115–128.

    Google Scholar 

  19. S. E. Payne, A tensor product action on q-clan generalized quadrangles with q = 2e, Linear Algebra Appl., Vol. 226/228 (1995) pp. 115–137.

    Google Scholar 

  20. S. E. Payne, The Subiaco notebook, http://www-math.cudenver.edu/spayne/publications/nroot.ps.

  21. S. E. Payne and C. Maneri, A family of skew-translation generalized quadrangles of even order, Congr. Numer., Vol. 36 (1982) pp. 127–135.

    Google Scholar 

  22. S. E. Payne and J. A. Thas, Finite Generalized Quadrangles, Pitman, London (1984).

    Google Scholar 

  23. T. Penttila, A plane representation of ovoids, Util. Math., Vol. 56 (1999) pp. 245–250.

    Google Scholar 

  24. T. Penttila and C. E. Praeger, Ovoids and translation ovals, J. London Math. Soc. (2), Vol. 56, No. 3 (1997) pp. 607–624.

    Google Scholar 

  25. I. Pinneri, Flocks, Generalised Quadrangles and Hyperovals, Ph.D. Thesis, University of Western Australia (1996).

  26. J. A. Thas, Ovoidal translation planes, Arch. Math., Vol. 23 (1972) pp. 110–112.

    Google Scholar 

  27. J. A. Thas, A remark concerning the restriction on the parameters of a 4-gonal subconfiguration, Simon Stevin, Vol. 48 (1974-75) pp. 65–68.

    Google Scholar 

  28. J. A. Thas, Generalized quadrangles and flocks of cones, European J. Combin., Vol. 8, No. 4 (1987) pp. 441–452.

    Google Scholar 

  29. J. A. Thas, A result on spreads of the generalized quadrangle T 2(O)with (O) an oval arising from a flock, and applications, European J. Combin., Vol. 22, No. 6 (2001) pp. 879–886.

    Google Scholar 

  30. J. Tits, Ovoides et groupes du Suzuki, Arch. Math., Vol. 13 (1962) pp. 187–198.

    Google Scholar 

  31. H. Van Maldeghem, Ovoids and spreads arising from involutions, Groups and Geometries, Trends in Mathematics, Birkäuser, Basel (1998) pp. 231–236.

    Google Scholar 

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Authors and Affiliations

  1. School of Mathematical Sciences, University of Adelaide, SA, 5005, Australia

    Matthew R. Brown & Christine M. O'Keefe

  2. Department of Mathematics, University of Colorado at Denver, Campus Box 170, P.O. Box 173364, Denver, CO, 80217-3364

    S. E. Payne

  3. Department of Mathematics, University of Western Australia, Crawley, WA, 6009, Australia

    Tim Penttila

  4. Department of Computer Science and Software Engineering, University of Western Australia, Crawley, WA, 6009, Australia

    Gordon F. Royle

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  1. Matthew R. Brown
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  2. Christine M. O'Keefe
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  4. Tim Penttila
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  5. Gordon F. Royle
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Brown, M.R., O'Keefe, C.M., Payne, S.E. et al. Spreads of T 2(o), α-flocks and Ovals. Designs, Codes and Cryptography 31, 251–282 (2004). https://doi.org/10.1023/B:DESI.0000015887.35146.14

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  • DOI: https://doi.org/10.1023/B:DESI.0000015887.35146.14

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