Designs, Codes and Cryptography

, Volume 9, Issue 1, pp 71–77 | Cite as

Hadamard matrices and dihedral groups

  • Hiroshi Kimura
Article
Received:
Accepted:
  • 27 Downloads

Abstract

Let D2p be a dihedral group of order 2p, where p is an odd integer. Let ZD2p be the group ring of D2p over the ring Z of integers. We identify elements of ZD2p and their matrices of the regular representation of ZD2p. Recently we characterized the Hadamard matrices of order 28 ([6] and [7]). There are exactly 487 Hadamard matrices of order 28, up to equivalence. In these matrices there exist matrices with some interesting properties. That is, these are constructed by elements of ZD6. We discuss relation of ZD2p and Hadamard matrices of order n=8p+4, and give some examples of Hadamard matrices constructed by dihedral groups.

Keywords

Data Structure Information Theory Interesting Property Discrete Geometry Group Ring 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
This is a preview of subscription content, log in to check access.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    M. Hall, Jr., Combinatorial Theory, Ginn (Blaisdell) Boston (1967).Google Scholar
  2. 2.
    M. Hall, Jr., Hadamard matrices of order 16, J. P. L. Research Summary, 36–10, Vol. 1 (1961) 21–26.Google Scholar
  3. 3.
    M. Hall, Jr., Hadamard matrices of order 20, J. P. L. Technical Report, (1965) 32–761.Google Scholar
  4. 4.
    N. Ito, J. S. Leon, and J. Q. Longyear, Classification of 3-(24,12,5) designs and 24-dimensional Hadamard matrices, J. Combin. Theory(A), Vol. 27 (1979) pp. 289–306.Google Scholar
  5. 5.
    H. Kimura, New Hadamard matrix of order 24, Graphs and Combin., Vol. 5 (1989) pp. 236–242.Google Scholar
  6. 6.
    H. Kimura, Classification of Hadamard matrices of order 28 with Hall sets, Discrete Math., Vol. 128 (1994) pp. 257–268.Google Scholar
  7. 7.
    H. Kimura, Classification of Hadamard matrices of order 28, Discrete Math., Vol. 133 (1994) pp. 171–180.Google Scholar
  8. 8.
    H. Kimura, A list of Hadamard matrices of order 28, unpublished manuscript.Google Scholar

Copyright information

© Kluwer Academic Publishers 1996

Authors and Affiliations

  • Hiroshi Kimura
    • 1
  1. 1.Department of MathematicsEhime UniversityMatsuyamaJapan

Personalised recommendations