Some large partial ovoids of \(Q^-(5,q)\), for odd \(q\)
- 139 Downloads
We give explicit descriptions of some of the largest partial ovoids of \(Q^-(5,q)\) currently known (for \(q\) odd). Apart from two generic constructions, the other (sporadic) examples are for fields of order \(q \le 11\). As far as we know, most of these results are new. Others were found earlier by computer but have never been described explicitly. All our constructions are formulated using elliptic quadratic forms that are not in standard form.
KeywordsPartial ovoid Elliptic quadric Generalized quadrangle
Mathematics Subject Classification05B25 51D20 51E12 51E20
I would like to thank Jan De Beule en Klaus Metsch for many valuable discussions and for introducing me to this subject. I would also like to thank the anonymous referees for many useful comments and suggestions.
- 1.Aguglia A., Cossidente A., Ebert G.L.: On pairs of permutable Hermitian surfaces. Discret. Math. 301(1), 28–33 (2005).Google Scholar
- 2.Brouwer A.: The \(O_6^-\)(3)-graph. http://www.win.tue.nl/~aeb/graphs/U4_3.html. Accessed 2013.
- 3.Cimráková M.: Search Algorithms for Substructures in Generalized Quadrangles. PhD Thesis, Ghent University (2005).Google Scholar
- 4.Cimráková M., Fack V.: Searching for maximal partial ovoids and spreads in generalized quadrangles. Bull. Belgian Math. Soc. 12(5), 697–705 (2005).Google Scholar
- 5.Coolsaet K., De Beule J., Siciliano A.: The known maximal partial ovoids of size \(q^2-1\) of \(Q(4, q)\). J. Comb. Des. 21(3), 89–100 (2013).Google Scholar
- 6.Cossidente A.: Some constructions on the Hermitian surface. Des. Codes Cryptogr. 51(2), 123–129 (2008).Google Scholar
- 7.De Beule J., Klein A., Metsch K.: Substructures of finite classical polar spaces. In: De Beule J., Storme L. (eds.) Current Research Topics in Galois Geometry. Nova Science Publishers, New York (2012).Google Scholar
- 8.De Beule J., Klein A., Metsch K., Storme L.: Partial ovoids and partial spreads in Hermitian polar spaces. Des. Codes Cryptogr. 47(1–3), 21–34 (2008).Google Scholar
- 9.Dye R.: Maximal sets of non-planar points of quadrics and symplectic polarities over gf(2). Geom. Dedicata 44, 281–293 (1992).Google Scholar
- 10.Ebert G., Hirschfeld J.: Complete systems of lines on a Hermitian surface over a finite field. Des. Codes Cryptogr. 17, 253–268 (1999).Google Scholar
- 11.Hirschfeld J.: Projective Geometries over Finite Fields, 2nd edn. Oxford Mathematical Monographs. Clarendon Press, Oxford (1998).Google Scholar
- 12.Hirschfeld J., Thas J.: General Galois Geometries. Oxford Mathematical Monographs. Clarendon Press, Oxford (1991).Google Scholar
- 13.Payne S., Thas J.: Finite Generalized Quadrangles. Pitman Research Notes in Mathematics Series 110. Longman, Boston (1984).Google Scholar