Designs, Codes and Cryptography

, Volume 58, Issue 1, pp 73–88 | Cite as

The cocyclic Hadamard matrices of order less than 40



In this paper all cocyclic Hadamard matrices of order less than 40 are classified. That is, all such Hadamard matrices are explicitly constructed, up to Hadamard equivalence. This represents a significant extension and completion of work by de Launey and Ito. The theory of cocyclic development is discussed, and an algorithm for determining whether a given Hadamard matrix is cocyclic is described. Since all Hadamard matrices of order at most 28 have been classified, this algorithm suffices to classify cocyclic Hadamard matrices of order at most 28. Not even the total numbers of Hadamard matrices of orders 32 and 36 are known. Thus we use a different method to construct all cocyclic Hadamard matrices at these orders. A result of de Launey, Flannery and Horadam on the relationship between cocyclic Hadamard matrices and relative difference sets is used in the classification of cocyclic Hadamard matrices of orders 32 and 36. This is achieved through a complete enumeration and construction of (4t, 2, 4t, 2t)-relative difference sets in the groups of orders 64 and 72.


Cocyclic Hadamard matrices Relative difference sets Classification of Hadamard matrices 

Mathematics Subject Classification (2000)

05B10 05B20 


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  1. 1.
    Bosma W., Cannon J., Playoust C.: The Magma algebra system. I. The user language. J. Symbolic Comput. 24, 235–265 (1997)MATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    de Launey W., Flannery D.L., Horadam K.J.: Cocyclic Hadamard matrices and difference sets. Discrete Appl. Math. 102(1–2), 47–61 (2000)MATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Flannery D.L.: Cocyclic Hadamard matrices and Hadamard groups are equivalent. J. Algebra 192(2), 749–779 (1997)MATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Horadam K.J.: Hadamard Matrices and their Applications. Princeton University Press, Princeton, NJ (2007)MATHGoogle Scholar
  5. 5.
    Horadam K.J., de Launey W.: Cocyclic development of designs. J. Algebraic Combin. 2(3), 267–290 (1993)MATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Ito N.: On Hadamard groups. J. Algebra 168(3), 981–987 (1994)MATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Ito N., Okamoto T.: On Hadamard groups of order 72. Algebra Colloq. 3(4), 307–324 (1996)MATHMathSciNetGoogle Scholar
  8. 8.
    Kharaghani H., Tayfeh-Rezaie B.: A Hadamard matrix of order 428. J. Combin. Des. 13(6), 435–440 (2005)MATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    McKay B.: Nauty User’s Guide, Version 2.2 (2007).
  10. 10.
    Ó Catháin P.: Group Actions on Hadamard matrices. M.Litt. Thesis, National University of Ireland, Galway (2008).
  11. 11.
    Orrick W.P.: Switching operations for Hadamard matrices. SIAM J. Discrete Math. 22(1), 31–50 (2008)MATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Röder M.: Quasiregular Projective Planes of Order 16—A Computational Approach. PhD thesis, Technische Universität Kaiserslautern (2006).
  13. 13.
    Röder M.: The quasiregular projective planes of order 16. Glasnik Matematicki. 43(2), 231–242 (2008)MATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Röder M.: RDS, Version 1.1 (2008).
  15. 15.
    Spence E.: Classification of Hadamard matrices of order 24 and 28. Discrete Math. 140(1–3), 185–243 (1995)MATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    The GAP Group.: GAP—Groups, Algorithms, and Programming, Version 4.4.12 (2008).

Copyright information

© Springer Science+Business Media, LLC 2010

Authors and Affiliations

  1. 1.School of Mathematics, Statistics and Applied MathematicsNational University of IrelandGalwayIreland

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