Small tight sets in finite elliptic, parabolic and Hermitian polar spaces

Abstract

Small tight sets have been classified in finite classical polar spaces of symplectic type, hyperbolic type and one of the two Hermitian types [1]. The approach used is not working for the remaining finite classical polar spaces, mainly due to the fact that a tight set for these is not necessarily a minihyper, so the known results on blocking sets can not be applied. This might be the reason that there is no classification for small tight sets in these spaces. This paper provides a classification of tight sets with parameter x in these spaces provided that x is small compared to the order q of the polar space. One of the results is that a tight set of the generalized quadrangle Q(4,q) that is not the union of disjoint lines has at least (\(\sqrt q \) + 1)(q + 1) points with equality if and only if it consists of the points of an embedded subquadrangle of order \(\sqrt q \).

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References

  1. [1]

    J. Bamberg, S. Kelly, M. Law and T. Penttila: Tight sets and m-ovoids of finite polar spaces, J. Combin. Theory Ser. 114 2007, 1293–1314.

    MathSciNet  Article  MATH  Google Scholar 

  2. [2]

    L. Beukemann and K. Metsch: Small tight sets of hyperbolic quadrics, Des. Codes Cryptogr. 68 2013, 11–24.

    MathSciNet  Article  MATH  Google Scholar 

  3. [3]

    P. J. Cameron and R. A. Liebler: Tactical decompositions and orbits of projective groups, Linear Algebra Appl. 46 1982, 91–102.

    MathSciNet  Article  MATH  Google Scholar 

  4. [4]

    J. DE Beule, A. Hallez and L. Storme: A non-existence result on Cameron-Liebler line classes, J. Combin. Des. 16 2008, 342–349.

    MathSciNet  Article  MATH  Google Scholar 

  5. [5]

    J. DE Beule, P. Govaerts, A. Hallez and L. Storme: Tight sets, weighted mcovers, weighted m-ovoids, and minihypers, Des. Codes Cryptogr. 50 2009, 187–201.

    MathSciNet  Article  MATH  Google Scholar 

  6. [6]

    B. DE Bruyn: On some 2-tight sets of polar spaces, tArs Combinatoria, submitted.

  7. [7]

    K. W. Drudge: Extremal sets in projective and polar spaces, ProQuest Llc, Ann Arbor, MI, 1998, Thesis (Ph.D.)-The University of Western Ontario (Canada).

    Google Scholar 

  8. [8]

    A. L. Gavrilyuk and I. Mogilnykh: Cameron-Liebler line classes in PG(n,4), Des. Codes Cryptogr., (to appear).

  9. [9]

    P. Govaerts and L. Storme: On Cameron-Liebler line classes, Adv. Geom. 4 2004, 279–286.

    MathSciNet  Article  MATH  Google Scholar 

  10. [10]

    J. W. P. Hirschfeld: Projective geometries over finite fields, Oxford Mathematical Monographs. The Clarendon Press Oxford University Press, New York, second edition, 1998.

    Google Scholar 

  11. [11]

    K. Metsch: The non-existence of Cameron-Liebler line classes with parameter 2 < x ≤ q, Bull. Lond. Math. Soc. 42 2010, 991–996.

    MathSciNet  Article  MATH  Google Scholar 

  12. [12]

    K. Metsch: An improved bound on the existence of Cameron-Liebler line classes, J. Combin. Theory Ser. 121 2014, 89–93.

    MathSciNet  Article  MATH  Google Scholar 

  13. [13]

    S. E. Payne: Tight pointsets in finite generalized quadrangles, Congr. Numer. 60 1987, 243–260. Eighteenth Southeastern International Conference on Combinatorics, Graph Theory, and Computing (Boca Raton, Fla., 1987).

    MathSciNet  MATH  Google Scholar 

  14. [14]

    S. E. Payne: Tight pointsets in finite generalized quadrangles. II, in: Proceedings of the Twenty-first Southeastern Conference on Combinatorics, Graph Theory, and Computing (Boca Raton, FL, 1990), volume 77, 31–41, 1990.

    MathSciNet  MATH  Google Scholar 

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Correspondence to Klaus Metsch.

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Metsch, K. Small tight sets in finite elliptic, parabolic and Hermitian polar spaces. Combinatorica 36, 725–744 (2016). https://doi.org/10.1007/s00493-015-3260-2

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Mathematics Subject Classication (2000)

  • 51A50
  • 51E20
  • 05B25