The Impact of Number Theory and Computer-Aided Mathematics on Solving the Hadamard Matrix Conjecture

Conference paper
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 43)


The Hadamard conjecture has been studied since the pioneering paper of Sylvester, “Thoughts on inverse orthogonal matrices, simultaneous sign successions, tessellated pavements in two or more colours, with applications to Newtons rule, ornamental tile work and the theory of numbers,” Phil Mag, 34 (1867), 461–475, first appeared. We review the importance of primes on those occasions that the conjecture, that for every odd number, t there exists an Hadamard matrix of order 4t – is confirmed. Although substantial advances have been made into the question of the density of this odd number, t it has still not been shown to have positive density. We survey the results of some computer-aided construction algorithms for Hadamard matrices.

Key words

Hadamard matrix Skew-Hadamard matrix Symmetric Hadamard matrix Asymptotic Hadamard matrices Prime number theorem Extended Riemann hypothesis Computer-aided construction 



The author sincerely thanks Professor Igor Shparlinski and Dr Daniel Gordon for alerting her to her confusion in Sect. 3.2 and pointing out pertinent references.


  1. 1.
    S.S. Agayan, Hadamard Matrices and Their Applications. Lecture Notes in Mathematics, vol 1168 (Springer, New York, 1985)Google Scholar
  2. 2.
    R. Craigen, Signed groups, sequences and the asymptotic existence of Hadamard matrices. J. Combin. Theory 71, 241–254 (1995)MathSciNetMATHCrossRefGoogle Scholar
  3. 3.
    R. Craigen, W.H. Holzmann, H. Kharaghani, On the asymptotic existence of complex Hadamard matrices. J. Combin. Des. 5, 319–327 (1997)MathSciNetMATHCrossRefGoogle Scholar
  4. 4.
    R. Craigen, J. Seberry, X.-M. Zhang, Product of four Hadamard matrices. J. Combin. Theory (Ser. A) 59, 318–320 (1992)MathSciNetMATHCrossRefGoogle Scholar
  5. 5.
    W. de Launey, On the asymptotic existence of partial complex Hadamard matrices and related combinatorial objects, in Conference on Coding, Cryptography and Computer Security, Lethbridge, AB, 1998. Discrete Appl. Math. 102, 37–45 (2000)MathSciNetMATHCrossRefGoogle Scholar
  6. 6.
    W. de Launey, A product for twelve Hadamard matrices. Aust. J. Combin. 7, 123–127 (1993)MATHGoogle Scholar
  7. 7.
    W. de Launey, On the asymptotic existence of Hadamard matrices. J. Combin. Theory (Ser. A) 116, 1002–1008 (2009)MathSciNetMATHCrossRefGoogle Scholar
  8. 8.
    W. de Launey, J. Dawson, An asymptotic result on the existence of generalised Hadamard matrices. J. Combin. Theory (Ser. A) 65, 158–163 (1994)MathSciNetMATHCrossRefGoogle Scholar
  9. 9.
    W. de Launey, D.M. Gordon, A comment on the Hadamard conjecture. J. Combin. Theory (Ser. A) 95, 180–184 (2001)MathSciNetMATHCrossRefGoogle Scholar
  10. 10.
    W. de Launey, D.M. Gordon, On the density of the set of known Hadamard orders. Cryptogr. Commun. 2, 233–246 (2010)MathSciNetMATHCrossRefGoogle Scholar
  11. 11.
    W. de Launey, D.M. Gordon, Should we believe the Hadamard conjecture?, in Conference in Honor of Warwick de Launey, IDA/CCR, La Jolla, CA, 16 May 2011Google Scholar
  12. 12.
    W. de Launey, H. Kharaghani, On the asymptotic existence of cocyclic Hadamard matrices. J. Combin. Theory (Ser. A) 116, 1140–1153 (2009)MathSciNetMATHCrossRefGoogle Scholar
  13. 13.
    W. de Launey, D.A Levin, A Fourier analytic approach to counting partial Hadamard matrices (21 March 2010). arXiv:1003.4003v1 [math.CO]Google Scholar
  14. 14.
    W. de Launey, M. Smith, Cocyclic orthogonal designs and the asymptotic existence of cocyclic Hadamard matrices and maximal size relative difference sets with forbidden subgroup of size 2. J. Combin. Theory (Ser. A) 93, 37–92 (2001)MathSciNetMATHCrossRefGoogle Scholar
  15. 15.
    D.Z Djokovic, Williamson matrices of order 4n of order n = 33, 35, 39. Discrete Math. 115, 267–271 (1993)MathSciNetMATHCrossRefGoogle Scholar
  16. 16.
    A.V. Geramita, J. Seberry, Orthogonal Designs: Quadratic Forms and Hadamard Matrices (Marcel Dekker, Boston, 1969)Google Scholar
  17. 17.
    G.M’. Edmonson, J. Seberry, M. Anderson, On the existence of Turyn sequences of length less than 43. Math. Comput. 62, 351–362 (1994)Google Scholar
  18. 18.
    S.W. Graham, I.E. Shparlinski, On RSA moduli with almost half of the bits prescribed. Discrete Appl. Math. 156, 3150–3154 (2008)MathSciNetMATHCrossRefGoogle Scholar
  19. 19.
    M. Hall Jr., Combinatorial Theory, 2nd edn. (Wiley, New York, 1986)MATHGoogle Scholar
  20. 20.
    W.H. Holzmann, H. Kharaghani, B. Tayfeh-Rezaie, Williamson matrices up to order 59. Designs Codes Cryptogr. 46, 343–352 (2008)MathSciNetMATHCrossRefGoogle Scholar
  21. 21.
    K.J. Horadam, Hadamard Matrices and Their Applications (Princeton University Press, Princeton, 2007)MATHGoogle Scholar
  22. 22.
    I. Livinski, Asymptotic existence of hadamard matrices. Master of Science Thesis, University of Manitoba, 2012Google Scholar
  23. 23.
    J. Horton, C. Koukouvinos, J. Seberry, A search for Hadamard matrices constructed from Williamson matrices. Bull. Inst. Combin. Appl. 35, 75–88 (2002)MathSciNetMATHGoogle Scholar
  24. 24.
    R.E.A.C. Paley, On orthogonal matrices. J. Math. Phys. 12, 311–320 (1933)Google Scholar
  25. 25.
    J. Seberry Wallis, On Hadamard matrices. J. Combin. Theory (Ser. A) 18, 149–164 (1975)MATHCrossRefGoogle Scholar
  26. 26.
    J. Seberry, M. Yamada, Hadamard matrices, sequences, and block designs, in Contemporary Design Theory: A Collection of Surveys, ed. by J.H. Dinitz, D.R. Stinson (Wiley, New York, 1992), pp. 431–560Google Scholar
  27. 27.
    J.J. Sylvester, Thoughts on inverse orthogonal matrices, simultaneous sign successions, tessellated pavements in two or more colours, with applications to Newton’s rule, ornamental tile work and the theory of numbers. Phil. Mag. 34, 461–475 (1867)Google Scholar
  28. 28.
    R. Turyn, An infinite class of Hadamard matrices. J. Combin. Theory (Ser. A) 12, 319–321 (1972)MathSciNetMATHCrossRefGoogle Scholar
  29. 29.
    J. Seberry Wallis, Hadamard matrices, in Combinatorics: Room Squares, Sum-Free Sets and Hadamard Matrices. Lecture Notes in Mathematics, ed. by W.D. Wallis, A. Penfold Street, J. Seberry Wallis (Springer, Berlin, 1972)Google Scholar
  30. 30.
    J. Williamson, Hadamard’s determinant theorem and the sum of four squares. Duke Math. J. 11, 65–81 (1944)MathSciNetMATHCrossRefGoogle Scholar
  31. 31.
    A.L. Whiteman, An infinite family of Hadamard matrices of Williamson type. J. Combin. Theory (Ser. A) 14, 334–340 (1972)MathSciNetCrossRefGoogle Scholar
  32. 32.
    M.Y. Xia, Some infinite families of Williamson matrices and difference sets. J. Combin. Theory (Ser. A) 61, 230–242 (1992)MathSciNetMATHCrossRefGoogle Scholar
  33. 33.
    M.Y. Xia, T.B. Xia, J. Seberry, G.X. Zuo, A new method for constructing T-matrices. Aust. J. Combin. 32, 61–78 (2005)MathSciNetMATHGoogle Scholar
  34. 34.
    M.Y. Xia, T. Xia, J. Seberry, A new method for constructing Williamson matrices. Designs Codes Cryptogr. 35, 191–209 (2005)MathSciNetMATHCrossRefGoogle Scholar
  35. 35.
    M. Yamada, On the Williamson type j matrices of orders 4 ⋅29, 4 ⋅41 and 4 ⋅37. J. Combin. Theory (Ser. A) 27, 378–381 (1979)MathSciNetMATHCrossRefGoogle Scholar
  36. 36.
    M. Yamada, Some new series of Hadamard matrices. Proc. Japan Acad. (Ser. A) 63, 86–89 (1987)MATHCrossRefGoogle Scholar
  37. 37.
    K. Yamamoto, M. Yamada, Williamson Hadamard matrices and Gauss sums. J. Math. Soc. Japan 37, 703–717 (1985)MathSciNetMATHCrossRefGoogle Scholar
  38. 38.
    K. Yamamoto, On congruences arising from relative Gauss sums. Number Theory and Combinatorics Japan 1984 (Tokyo, Okayama and Kyoto, 1984). World Scientific Publ., Singapore, pp. 423–446 (1985)Google Scholar
  39. 39.
    G.X. Zuo, M.Y. Xia, A special class of T-matrices. Designs Codes Cryptogr. 54, 21–28 (2010)MathSciNetMATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  1. 1.Centre for Computer and Information Security Research, SCSSEUniversity of WollongongWollongongAustralia

Personalised recommendations