Cameron-Liebler k-classes in PG(2k+1, q)

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Abstract

We look at a generalization of Cameron-Liebler line classes to sets of k-spaces, focusing on results in PG(2k+1, q). Here we obtain a connection to k-spreads which parallels the situation for line classes in PG(3,q). After looking at some characterizations of these sets and some of the difficulties that arise in contrast to the known results for line classes, we give some connections to various other geometric objects including k-spreads and Erdős–Ko–Rado sets, and prove results concerning the existence of these objects.

Mathematics Subject Classification (2000)

51E20 05B25 05E30 

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References

  1. [1]
    L. Beukemann and K. Metsch: Small tight sets of hyperbolic quadrics, Des. Codes Cryptogr. 68 (2013), 11–24.MathSciNetCrossRefMATHGoogle Scholar
  2. [2]
    A. Blokhuis, A. E. Brouwer, A. Chowdhury, P. Frankl, T. Mussche, B. Patkós and T. Szőnyi: A Hilton–Milner theorem for vector spaces, Electronic Journal of Combinatorics 17 (2010), R71.MathSciNetMATHGoogle Scholar
  3. [3]
    A. Blokhuis, A. E. Brouwer and T. Szőnyi: On the chromatic number of q-Kneser graphs, Designs, Codes and Cryptography 65 (2012), 187–197.MathSciNetCrossRefMATHGoogle Scholar
  4. [4]
    R.C. Bose: A note on Fisher’s inequality for balanced incomplete block designs, The Annals of Mathematical Statistics 20 (1949), 619–620.MathSciNetCrossRefMATHGoogle Scholar
  5. [5]
    A. A. Bruen and K. Drudge: The construction of Cameron–Liebler line classes in PG(3,q), Finite Fields and Their Applications 5 (1999), 35–45.MathSciNetCrossRefMATHGoogle Scholar
  6. [6]
    P. J. Cameron and R. A. Liebler: Tactical decompositions and orbits of projective groups, Linear Algebra and its Applications 46 (1982), 91–102.MathSciNetCrossRefMATHGoogle Scholar
  7. [7]
    J. De Beule, J. Demeyer, K. Metsch and M. Rodgers: A new family of tight sets in Q+(5;q), Designs, Codes and Cryptography (DOI: 10.1007/s10623-014-00239), 2014.Google Scholar
  8. [8]
    J. De Beule, P. Govaerts, A. Hallez and L. Storme: Tight sets, weighted m-covers, weighted m-ovoids, and minihypers, Designs, Codes and Cryptography 50 (2009), 187–201.MathSciNetCrossRefMATHGoogle Scholar
  9. [9]
    J. De Beule, A. Hallez and L. Storme: A non-existence result on Cameron–Liebler line classes, Journal of Combinatorial Designs 16 (2008), 342–349.MathSciNetCrossRefMATHGoogle Scholar
  10. [10]
    M. De Boeck: Intersection problems in finite geometries, PhD thesis, Ghent University, 2014.Google Scholar
  11. [11]
    P. Delsarte: Association schemes and t-designs in regular semilattices, Journal of Combinatorial Theory, Series A 20 (1976), 230–243.MathSciNetCrossRefMATHGoogle Scholar
  12. [12]
    P. Dembowski: Finite Geometries, Springer-Verlag, Berlin, 1968.CrossRefMATHGoogle Scholar
  13. [13]
    K. Drudge: On a conjecture of Cameron and Liebler, European Journal of Combinatorics 20 (1999), 263–269.MathSciNetCrossRefMATHGoogle Scholar
  14. [14]
    T. Feng, K. Momihara and Q. Xiang: Cameron–Liebler line classes with parameter x= q 2−1/2, Journal of Combinatorial Theory, Series A 133 (2015), 307–338.MathSciNetCrossRefMATHGoogle Scholar
  15. [15]
    P. Frankl and R. M. Wilson: The Erdős–Ko–Rado theorem for vector spaces, Journal of Combinatorial Theory, Series A 43 (1986), 228–236.MathSciNetCrossRefMATHGoogle Scholar
  16. [16]
    A. Gavrilyuk and K. Metsch: A modular equality for Cameron–Liebler line classes, Journal of Combinatorial Theory, Series A 127 (2014), 224–242.MathSciNetCrossRefMATHGoogle Scholar
  17. [17]
    P. Govaerts and T. Penttila: Cameron–Liebler line classes in PG(3; 4), Bulletin of the Belgian Mathematical Society–Simon Stevin 12 (2005), 793–804.MathSciNetMATHGoogle Scholar
  18. [18]
    P. Govaerts and L. Storme: On Cameron–Liebler line classes, Advances in Geometry 4 (2004), 279–286.MathSciNetCrossRefMATHGoogle Scholar
  19. [19]
    Inc. Gurobi Optimization, Gurobi optimizer reference manual, 2014.Google Scholar
  20. [20]
    J. W. P. Hirschfeld: The double-six of lines over PG(3; 4), Journal of the Australian Mathematical Society 4 (1964), 83–89.MathSciNetCrossRefMATHGoogle Scholar
  21. [21]
    A. Klein, K. Metsch and L. Storme: Small maximal partial spreads in classical finite polar spaces, Advances in Geometry 10 (2010), 379–402.MathSciNetCrossRefMATHGoogle Scholar
  22. [22]
    B. Lv and K. Wang: The eigenvalues of q-Kneser graphs, Discrete Mathematics 312 (2012), 1144–1147.MathSciNetCrossRefMATHGoogle Scholar
  23. [23]
    K. Metsch: The non-existence of Cameron–Liebler line classes with parameter 2<xq, Bulletin of the London Mathematical Society 42 (2010), 991–996.MathSciNetCrossRefMATHGoogle Scholar
  24. [24]
    K. Metsch: An improved bound on the existence of Cameron–Liebler line classes, J. Combin. Theory Ser. A 121 (2014), 89–93.MathSciNetCrossRefMATHGoogle Scholar
  25. [25]
    M. W. Newman: Independent sets and eigenspaces, PhD thesis, University of Waterloo, 2004.Google Scholar
  26. [26]
    T. G. Ostrom: Vector spaces and construction of finite projective planes, Archiv der Mathematik 19 (1968), 1–25.MathSciNetCrossRefMATHGoogle Scholar
  27. [27]
    T. Penttila: Cameron–Liebler line classes in PG(3,q), Geometriae Dedicata 37 (1991), 245–252.MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© János Bolyai Mathematical Society and Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  • Morgan Rodgers
    • 1
  • Leo Storme
    • 2
  • Andries Vansweevelt
    • 3
  1. 1.School of Mathematics and Computer ScienceLake Superior State UniversitySault Sainte MarieUSA
  2. 2.Department of MathematicsGhent UniversityGhentBelgium
  3. 3.WiskundeKU Leuven Kulak KortrijkKortrijkBelgium

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