Advertisement

Designs, Codes and Cryptography

, Volume 86, Issue 5, pp 997–1006 | Cite as

On m-ovoids of regular near polygons

  • John Bamberg
  • Jesse Lansdown
  • Melissa Lee
Article
  • 105 Downloads

Abstract

We generalise the work of Segre (Ann Mat Pura Appl 4(70):1–201, 1965), Cameron et al. (J Algebra 55(2):257–280, 1978), and Vanhove (J Algebr Comb 34(3):357–373, 2011) by showing that nontrivial m-ovoids of the dual polar spaces \(\mathsf {DQ}(2d, q)\), \(\mathsf {DW}(2d-1,q)\) and \(\mathsf {DH}(2d-1,q^2)\) (\(d\geqslant 3\)) are hemisystems. We also provide a more general result that holds for regular near polygons.

Keywords

Regular near polygon Dual polar space Hemisystem 

Mathematics Subject Classification

05B25 51E12 51E20 

Notes

Acknowledgements

The first author acknowledges the support of the Australian Research Council (ARC) Future Fellowship FT120100036. The second author acknowledges the support of an Australian Postgraduate Award and a UWA Top-Up Scholarship. The third author acknowledges the support of a Hackett Postgraduate Research Scholarship.

References

  1. 1.
    Bamberg J., Betten A., Cara P., De Beule J., Lavrauw M., Neunhöffer M.: FinInG—Finite Incidence Geometry, Version 1.0. (2014).Google Scholar
  2. 2.
    Brouwer A.E., Cohen A.M., Neumaier A.: Distance-Regular Graphs. Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 18. Springer, Berlin (1989).Google Scholar
  3. 3.
    Cameron P.J.: Dual polar spaces. Geom. Dedic. 12(1), 75–85 (1982).MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Cameron P.J., Goethals J.-M., Seidel J.J.: Strongly regular graphs having strongly regular subconstituents. J. Algebra 55(2), 257–280 (1978).MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Cooperstein B.N., Pasini A.: The non-existence of ovoids in the dual polar space \({{\rm DW}}(5,q)\). J. Comb. Theory A 104(2), 351–364 (2003).MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    De Bruyn B.: Near Polygons. Frontiers in Mathematics. Birkhäuser Verlag, Basel (2006).CrossRefGoogle Scholar
  7. 7.
    De Bruyn B.: Isometric full embeddings of \(DW(2n-1, q)\) into \(DH(2n-1, q^2)\). Finite Fields Appl. 14(1), 188–200 (2008).MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    De Bruyn B., Pralle H.: On small and large hyperplanes of \({\rm DW}(5, q)\). Graphs Comb. 23(4), 367–380 (2007).MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    De Bruyn B., Vanhove F.: Inequalities for regular near polygons, with applications to \(m\)-ovoids. Eur. J. Comb. 34(2), 522–538 (2013).MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Delsarte P.: Pairs of vectors in the space of an association scheme. Philips Res. Rep. 32(5–6), 373–411 (1977).MathSciNetGoogle Scholar
  11. 11.
    Gurobi Optimization, Inc.: Gurobi optimizer reference manual (2016).Google Scholar
  12. 12.
    Payne S.E., Thas J.A.: Finite Generalized Quadrangles. EMS Series of Lectures in Mathematics, 2nd edn. European Mathematical Society (EMS), Zürich (2009).CrossRefGoogle Scholar
  13. 13.
    Segre B.: Forme e geometrie hermitiane, con particolare riguardo al caso finito. Ann. Mat. Pura Appl. 4(70), 1–201 (1965).MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Shult E., Yanushka A.: Near \(n\)-gons and line systems. Geom. Dedic. 9(1), 1–72 (1980).MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    The GAP Group: GAP—Groups, Algorithms, and Programming, Version 4.8.6 (2016).Google Scholar
  16. 16.
    Thomas S.: Designs and partial geometries over finite fields. Geom. Dedic. 63(3), 247–253 (1996).MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Tits J.: Sur la trialité et certains groupes qui s’en déduisent. Inst. Hautes Études Sci. Publ. Math. 2, 13–60 (1959).CrossRefzbMATHGoogle Scholar
  18. 18.
    Vanhove F.: A Higman inequality for regular near polygons. J. Algebr. Comb. 34(3), 357–373 (2011).MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Vanhove F.: Incidence geometry from an algebraic graph theory point of view. PhD thesis, Ghent University (2011).Google Scholar

Copyright information

© Springer Science+Business Media, LLC 2017

Authors and Affiliations

  1. 1.Centre for the Mathematics of Symmetry of Computation, School of Mathematics and StatisticsUniversity of Western AustraliaPerthAustralia
  2. 2.Department of MathematicsImperial CollegeLondonUK

Personalised recommendations