Ovoidal blocking sets and maximal partial ovoids of Hermitian varieties | Designs, Codes and Cryptography
Skip to main content
Springer Nature Link
Log in

Ovoidal blocking sets and maximal partial ovoids of Hermitian varieties

  • Published:
Designs, Codes and Cryptography Aims and scope Submit manuscript

Abstract

In Mazzocca et al. (Des. Codes Cryptogr. 44:97–113, 2007), large minimal blocking sets in PG(3, q 2) and PG(4, q 2) have been constructed starting from ovoids of PG(3, q), Q(4, q) and Q(6, q). Some of these can be embedded in a Hermitian variety as maximal partial ovoids. In this paper, the geometric conditions assuring these embeddings are established.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
¥17,985 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price includes VAT (Japan)

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. André J.: Über nicht-Desarguessche Ebenen mit transitiver Translationsgruppe. Math. Z. 60, 156–186 (1954)

    Article  MATH  MathSciNet  Google Scholar 

  2. Ball S.: On ovoids of O(5, q). Adv. Geom. 4(1), 1–7 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  3. Ball S., Govaerts P., Storme L.: On ovoids of parabolic quadrics. Des. Codes Cryptogr. 38(1), 131–145 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  4. Barlotti A., Cofman J.: Finite sperner spaces constructed from projective and affine spaces. Abh. Math. Semin. Univ. Hamb. 40, 231–241 (1974)

    Article  MATH  MathSciNet  Google Scholar 

  5. Bruck R.H., Bose R.C.: The construction of translation planes from projective spaces. J. Algebra. 1, 85–102 (1964)

    Article  MATH  MathSciNet  Google Scholar 

  6. Bruen A.A., Thas J.A.: Hyperplane coverings and blocking Sets. Math. Z. 181, 407–409 (1982)

    Article  MATH  MathSciNet  Google Scholar 

  7. Buekenhout F.: Existence of unitals in finite translation planes of order q 2 with kernel of order q. Geom. Dedicata. 5, 189–194 (1976)

    Article  MATH  MathSciNet  Google Scholar 

  8. De Beule J., Klein A., Metsch K., Storme L.: Partial ovoids and partial spreads of classical finite polar spaces. Serdica Math. J. 34(4), 689–714 (2008)

    MathSciNet  Google Scholar 

  9. Kantor W.M.: Ovoids and translation planes. Canad. J. Math. 34, 1195–1207 (1982)

    MATH  MathSciNet  Google Scholar 

  10. Lunardon G.: Blocking sets and semifields. J. Comb. Theory A. 113(6), 1172–1188 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  11. Mazzocca F., Polverino O.: Blocking sets in PG(2, q n) from cones of PG(2n, q). J. Algebraic Combin. 24(1), 61–81 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  12. MazzoccaF. Polverino O., Storme L.: Blocking sets in PG(r, q n). Des. Codes Cryptogr. 44, 97–113 (2007)

    Article  MathSciNet  Google Scholar 

  13. Metsch K.: A note on Buekenhout–Metz unitals. In: J.W.P. Hirschfeld et al.(eds.) Geometry, combinatorial designs and related structures. Proceedings of the first Pythagorean conference, Island of Spetses, Greece, June 1–7, 1996. London Mathematical Society Lecture Note Series, vol. 245, pp. 177–180. Cambridge University Press, Cambridge (1997).

  14. Metz K.: On a class of unitals. Geom. Dedicata 8, 125–126 (1979)

    Article  MATH  MathSciNet  Google Scholar 

  15. Segre B.: Teoria di Galois, fibrazioni proiettive e geometrie non desarguesiane. Ann. Mat. Pura Appl. 64, 1–76 (1964)

    Article  MATH  MathSciNet  Google Scholar 

  16. Szőnyi T., Cossidente A., Gács A., Mengyán C., Siciliano A., Weiner Zs.: On large minimal blocking sets in PG(2, q). J. Comb. Des. 13, 25–41 (2005)

    Article  Google Scholar 

  17. Thas J.A.: Semi–partial geometries and spreads of classical polar spaces. J. Comb. Theory Ser. A. 35, 58–66 (1983)

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Dip. di Matematica, Seconda Università degli Studi di Napoli, 81100, Caserta, Italy

    G. Marino & O. Polverino

Authors
  1. G. Marino
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. O. Polverino
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to O. Polverino.

Additional information

Communicated by Leo Storme.

Dedicated to the memory of András Gács (1969–2009).

Rights and permissions

Reprints and permissions

About this article

Cite this article

Marino, G., Polverino, O. Ovoidal blocking sets and maximal partial ovoids of Hermitian varieties. Des. Codes Cryptogr. 56, 115–130 (2010). https://doi.org/10.1007/s10623-010-9390-z

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10623-010-9390-z

Keywords

Mathematics Subject Classification (2000)

Search

Navigation