Advertisement

Blocking sets of tangent and external lines to a hyperbolic quadric in \(\varvec{PG(3,q)}\), \(\varvec{q}\) even

  • Binod Kumar Sahoo
  • Bikramaditya SahuEmail author
Article
  • 37 Downloads

Abstract

Let \(\mathcal {H}\) be a fixed hyperbolic quadric in the three-dimensional projective space PG(3, q), where q is a power of 2. Let \(\mathbb {E}\) (respectively \(\mathbb {T}\)) denote the set of all lines of PG(3, q) which are external (respectively tangent) to \(\mathcal {H}\). We characterize the minimum size blocking sets of PG(3, q) with respect to each of the line sets \(\mathbb {T}\) and \(\mathbb {E}\cup \mathbb {T}\).

Keywords

Projective space blocking set irreducible conic hyperbolic quadric, generalized quadrangle ovoid 

2010 Mathematics Subject Classification

05B25 51E21 

Notes

Acknowledgements

The authors wish to thank Prof. Bart De Bruyn for his comments, which helped improve an earlier version of the article.

References

  1. 1.
    Aguglia A and Giulietti M, Blocking sets of certain line sets related to a conic, Des. Codes Cryptogr. 39 (2006) 397–405MathSciNetCrossRefGoogle Scholar
  2. 2.
    Barlotti A, Un’estensione del teorema di Segre-Kustaanheimo, Boll. Un. Mat. Ital. 10 (1955) 498–506MathSciNetzbMATHGoogle Scholar
  3. 3.
    Biondi P and Lo Re P M, On blocking sets of external lines to a hyperbolic quadric in \(PG(3, q)\), \(q\) even, J. Geom. 92 (2009) 23–27MathSciNetCrossRefGoogle Scholar
  4. 4.
    Biondi P, Lo Re P M and Storme L, On minimum size blocking sets of external lines to a quadric in \(PG(3, q)\), Beiträge Algebra Geom. 48 (2007) 209–215MathSciNetzbMATHGoogle Scholar
  5. 5.
    Bose R C and Burton R C, A characterization of flat spaces in a finite geometry and the uniqueness of the Hamming and the MacDonald codes, J. Comb. Theory1 (1966) 96–104MathSciNetCrossRefGoogle Scholar
  6. 6.
    Fellegara G, Gli ovaloidi in uno spazio tridimensionale di Galois di ordine 8, Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Nat. (8) 32 (1962) 170–176MathSciNetzbMATHGoogle Scholar
  7. 7.
    Giulietti M, Blocking sets of external lines to a conic in \(PG(2, q)\), \(q\) even, European J. Combin. 28 (2007) 36–42MathSciNetCrossRefGoogle Scholar
  8. 8.
    Hirschfeld J W P, Finite projective spaces of three dimensions (1985) (New York: Oxford University Press)zbMATHGoogle Scholar
  9. 9.
    Moorhouse G E, Incidence geometry (2017), available online at http://ericmoorhouse.org/handouts/Incidence_Geometry.pdf
  10. 10.
    O’Keefe C M and Penttila T, Ovoids of \(PG(3,16)\) are elliptic quadrics, J. Geom. 38 (1990) 95–106MathSciNetCrossRefGoogle Scholar
  11. 11.
    O’Keefe C M and Penttila T, Ovoids of \(PG(3,16)\) are elliptic quadrics, II, J. Geom. 44 (1992) 140–159MathSciNetCrossRefGoogle Scholar
  12. 12.
    O’Keefe C M, Penttila T and Royle G F, Classification of ovoids in \(PG(3,32)\), J. Geom. 50 (1994) 143–150MathSciNetCrossRefGoogle Scholar
  13. 13.
    Panella G, Caratterizzazione delle quadriche di uno spazio (tridimensionale) lineare sopra un corpo finito, Boll. Un. Mat. Ital. 10 (1955) 507–513MathSciNetzbMATHGoogle Scholar
  14. 14.
    Patra K L, Sahoo B K and Sahu B, Minimum size blocking sets of certain line sets related to a conic in \(PG(2, q)\), Discrete Math. 339 (2016) 1716–1721MathSciNetCrossRefGoogle Scholar
  15. 15.
    Payne S E and Thas J A, Finite generalized quadrangles (2009) (Zürich: European Mathematical Society)CrossRefGoogle Scholar
  16. 16.
    Sahoo B K and Sahu B, Blocking sets of certain line sets to a hyperbolic quadric in \(PG(3,q)\), Adv. Geom., to appear,  https://doi.org/10.1515/advgeom-2018-0009
  17. 17.
    Sahoo B K and Sastry N S N, Binary codes of the symplectic generalized quadrangle of even order, Des. Codes Cryptogr. 79 (2016) 163–170MathSciNetCrossRefGoogle Scholar
  18. 18.
    Segre B, On complete caps and ovaloids in three-dimensional Galois spaces of characteristic two, Acta Arith. 5 (1959) 315–332MathSciNetCrossRefGoogle Scholar
  19. 19.
    Thas J A, Ovoidal translation planes, Arch. Math. (Basel) 23 (1972) 110–112MathSciNetCrossRefGoogle Scholar

Copyright information

© Indian Academy of Sciences 2018

Authors and Affiliations

  1. 1.School of Mathematical SciencesNational Institute of Science Education and Research, Bhubaneswar, HBNIJatni, Khurda DistrictIndia

Personalised recommendations