A nonexistence result for abelian menon difference sets using perfect binary arrays

Abstract

A Menon difference set has the parameters (4N 2,2N 2-N, N 2-N). In the abelian case it is equivalent to a perfect binary array, which is a multi-dimensional matrix with elements ±1 such that all out-of-phase periodic autocorrelation coefficients are zero. Suppose that the abelian group\(H \times K \times Z_{p^\alpha }\) contains a Menon difference set, wherep is an odd prime, |K|=p α, andp j≡−1 (mod exp (H)) for somej. Using the viewpoint of perfect binary arrays we prove thatK must be cyclic. A corollary is that there exists a Menon difference set in the abelian group\(H \times K \times Z_{3^\alpha }\), where exp(H)=2 or 4 and |K|=3α, if and only ifK is cyclic.

This is a preview of subscription content, access via your institution.

References

  1. [1]

    K.T. Arasu, J.A. Davis, J. Jedwab, andS.K. Sehgal: New constructions of Menon difference sets,J. Comb. Theory (A),64 329–336, 1993.

    Google Scholar 

  2. [2]

    W.-K. Chan, S.-L. Ma, andM.-K. Siu: Non-existence of certain perfect arrays,Discrete Math. To appear.

  3. [3]

    W.-K. Chan, andM.-K. Siu: Summary of perfects×t arrays, 1≤st≤100,Electron. Lett.,27 709–710, 1991. (CorrectionElectron. Lett.27 1112, 1991).

    Google Scholar 

  4. [4]

    W.K. Chan: Perfect arrays and Menon difference sets, 1991, M. Phil. thesis.

  5. [5]

    J.A. Davis, andJ. Jedwab: A survey of Hadamard difference sets. Submitted.

  6. [6]

    J. Jedwab:Perfect arrays, Barker arrays and difference sets, PhD thesis, University of London, 1991.

  7. [7]

    J. Jedwab: Generalized perfect arrays and Menon difference sets,Designs, Codes and Cryptography,2 19–68, 1992.

    Google Scholar 

  8. [8]

    J. Jedwab, andJ.A. Davis: Nonexistence of certain perfect binary arrays,Electron. Lett.,29 99–101, 1993.

    Google Scholar 

  9. [9]

    L.E. Kopilovich: On perfect binary arrays,Electron. Lett.,24 566–567, 1988.

    Google Scholar 

  10. [10]

    E.S. Lander:Symmetric Designs: an Algebraic Approach, London Mathematical Society Lecture Notes Series74. Cambridge University Press, Cambridge, 1983.

    Google Scholar 

  11. [11]

    S.L. Ma: Polynomial addition sets and polynomial digraphs,Linear Algebra and its Applications,69 213–230, 1985.

    Google Scholar 

  12. [12]

    R.L. McFarland: Difference sets in abelian groups of order 4p 2,Mitt. Math. Sem. Giessen,192 1–70, 1989.

    Google Scholar 

  13. [13]

    R.L. McFarland: Necessary conditions for Hadamard difference sets, In D. RayChaudhuri, editor,The IMA Volumes in Mathematics and its Applications, Vol. 21 Coding Theory and Design Theory, 257–272. Springer-Verlag, New York, 1990.

    Google Scholar 

  14. [14]

    R.L. McFarland: Sub-difference sets of Hadamard difference sets,J. Comb. Theory (A),54, 112–122, 1990.

    Google Scholar 

  15. [15]

    R.J. Turyn: Character sums and difference sets,Pacific J. Math.,15 319–346, 1965.

    Google Scholar 

  16. [16]

    M.-Y. Xia: Some infinite classes of special Williamson matrices and difference sets,J. Comb. Theory (A),61 230–242, 1992.

    Google Scholar 

Download references

Author information

Affiliations

Authors

Additional information

This work is partially supported by NSA grant # MDA 904-92-H-3057 and by NSF grant # NCR-9200265. The author thanks the Mathematics Department, Royal Holloway College, University of London for its hospitality during the time of this research

This work is partially supported by NSA grant # MDA 904-92-H-3067

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Arasu, K.T., Davis, J.A. & Jedwab, J. A nonexistence result for abelian menon difference sets using perfect binary arrays. Combinatorica 15, 311–317 (1995). https://doi.org/10.1007/BF01299738

Download citation

Mathematics Subject Classification (1991)

  • 05 B 10
  • 20