Abstract
A generalization of the concept of a flock of a quadratic cone in PG(3,q), q even, where the base of the cone is replaced by a translation oval, was introduced in [4] and is the focus of this work. The related idea of a q-clan is also generalized and studied with a particular emphasis on the connections with hyperovals. Several examples are given leading to new proofs of the existence of known hyperovals, unifying much of what has been done in this area. Finally, a proof that the Cherowitzo hyperovals do form an infinite family is also included.
Similar content being viewed by others
References
Bader, L., Lunardon, G. and Payne, S. E.: On q-clan geometry, q = 2e, Bull. Belg. Math. Soc. Simon Stevin 1 (1994), 301–328.
Cherowitzo, W.: Hyperovals in Desarguesian planes of even order, Ann. Discrete Math. 37 (1988), 87–94.
Cherowitzo, W., Penttila, T., Pinneri, I. and Royle, G. F.: Flocks and ovals, Geom. Dedicata 60 (1996), 17–37.
Fisher, J. C. and Thas, J. A.: Flocks in PG(3; q), Math. Z. 169 (1979), 1–11.
Glynn, D. G.: Two new sequences of ovals in finite desarguesian planes of even order, in: L. R. A. Casse (ed.), Combinatorial Mathematics X (Adelaide 1982), Lecture Notes in Math. 1036, Springer, New York, 1983, pp. 217–229.
Glynn, D. G.: A condition for the existence of ovals in PG(2, q), q even, Geom. Dedicata 32 (1989), 247–252.
Hirschfeld, J. W. P.: Projective Geometries over Finite Fields, Oxford University Press, Oxford, 1979.
Korchmáros, G.: Old and new results on ovals in finite projective planes, in: Surveys in Combinatorics, 1991 (Guildford, 1991), London Math. Soc. Lecture Note Ser., 166, Cambridge Univ. Press, Cambridge, 1991, pp. 41–72.
O'Keefe, C. M. and Penttila, T.: Polynomials for hyperovals of desarguesian planes, J. Austral. Math. Soc. A 51 (1991), 436–447.
O'Keefe, C. M. and Penttila, T.: A new hyperoval in PG(2; 32), J. Geom. 44 (1992), 117–139.
O'Keefe, C. M., Penttila, T. and Praeger, C. E.: Stabilisers of hyperovals in PG(2; 32), in J. W. P. Hirschfeld et al. (eds.), Advances in Finite Geometries and Designs, Oxford University Press, Oxford, 1991, pp. 337–357.
O'Keefe, C. M. and Thas, J. A.: Collineations of Subiaco and Cherowitzo hyperovals, Bull. Belg. Math. Soc. 3 (1996), 179–193.
Payne, S. E.: A new infinite family of generalized quadrangles, Congr. Numer. 49 (1985), 115– 128.
Payne, S. E.: An essay on skew translation generalized quadrangles, Geom. Dedicata 32 (1989), 93–118.
Payne, S. E.: Collineations of the generalized quadrangles associated with q-clans, Ann. Discrete Math. 52 (1992), 449–461.
Payne, S. E.: The fundamental theorem of q-clan geometry, Des. Codes Cryptogr. 8 (1996), 181–202.
Payne, S. E. and Thas, J. A.: Conical flocks, partial flocks, derivation and generalized quadrangles, Geom. Dedicata 38 (1991), 229–243.
Segre, B.: Ovals in a finite projective plane, Canad. J. Math. 7 (1955), 414–416.
Thas, J. A.: Generalized quadrangles and flocks of cones, European J. Combin. 8 (1987), 441– 452.
Walker, M.: A class of translation planes, Geom. Dedicata 5 (1976), 135–146.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Cherowitzo, W. α-Flocks and Hyperovals. Geometriae Dedicata 72, 221–245 (1998). https://doi.org/10.1023/A:1005022808718
Issue Date:
DOI: https://doi.org/10.1023/A:1005022808718