Skip to main content
SpringerLink
Log in

Difference Sets and Hyperovals

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

Abstract

We construct three infinite families of cyclic difference sets, using monomial hyperovals in a desarguesian projective plane of even order. These difference sets give rise to cyclic Hadamard designs, which have the same parameters as the designs of points and hyperplanes of a projective geometry over the field with two elements. Moreover, they are substructures of the Hadamard design that one can associate with a hyperoval in a projective plane of even order.

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

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. R. Bacher, Cyclic difference sets with parameters (511,255,127), L'Ens. Math., Vol. 40 (1994) pp. 187-192.

    Google Scholar 

  2. L. D. Baumert, Cyclic difference sets, Lecture Notes in Mathematics, Springer, Berlin, 182 (1971).

  3. D. G. Glynn, Two new sequences of ovals in Desarguesian planes of even order, Combinatorial Mathematics X (L. R. A. Casse, ed.), Lecture Notes in Mathematics, Springer, Berlin, 14 (1982) pp. 217-229.

    Google Scholar 

  4. J. W. P. Hirschfeld, Ovals in a Desarguesian plane of even order, Annali Mat. Pura Appl., Vol. 102 (1975) pp. 79-89.

    Google Scholar 

  5. W.-A. Jackson, A characterisation of Hadamard designs with SL(2, q) acting transitively, Geom. Dedicata, Vol. 46 (1993) pp. 197-206.

    Google Scholar 

  6. W.-A. Jackson and P.R. Wild, On GMW designs and cyclic Hadamard designs, Designs, Codes and Cryptography, Vol. 10 (1997) pp. 185-191

    Google Scholar 

  7. D. Jungnickel, Difference sets, Contemporary design theory. A collection of surveys (J.H. Dinitz and D.R. Stinson, eds), Wiley, New York (1992) pp. 241-324.

    Google Scholar 

  8. G. Korchmaros, Old and new results on ovals of finite projective planes, Surveys in Combinatorics 1991 (A.D. Keedwell, ed.), London Math. Soc. Lecture Note Series 166 (1991) pp. 41-72.

  9. A. Maschietti, On Hadamard designs associated with a hyperoval, J. Geometry, Vol. 53 (1995) pp. 122-130.

    Google Scholar 

  10. T. E. Norwood and Q. Xiang, On GMW designs and a conjecture of Assmus and Key, J. Comb. Theory (A) (to appear).

  11. S. E. Payne, A complete determination of translation ovoids in finite desarguesian planes, Atti Accad. Naz. Lincei Rend., Vol. 51 (1971) pp. 328-331.

    Google Scholar 

  12. A. Pott, Finite geometry and character theory, Lecture Notes in Mathematics, Springer, Berlin, 1601 (1995).

    Google Scholar 

  13. B. Segre, Ovals in a finite projective plane, Canad. J. Math., Vol. 7 (1955) pp. 414-416.

    Google Scholar 

  14. B. Segre and U. Bartocci, Ovali ed altre curve nei piani di Galois di caratteristica due, Acta Arith., Vol. 8 (1971) pp. 423-449.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Dipartimento di Matematica G. Castelnuovo, Universita' La Sapienza, Roma.

    Antonio Maschietti

Authors
  1. Antonio Maschietti
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

About this article

Cite this article

Maschietti, A. Difference Sets and Hyperovals. Designs, Codes and Cryptography 14, 89–98 (1998). https://doi.org/10.1023/A:1008264606494

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1023/A:1008264606494

Search

Navigation