Designs, Codes and Cryptography

, Volume 64, Issue 1–2, pp 143–151

New restrictions on possible orders of circulant Hadamard matrices



We obtain several new number theoretic results which improve the field descent method. We use these results to rule out many of the known open cases of the circulant Hadamard matrix conjecture. In particular, the only known open case of the Barker sequence conjecture is settled.


Circulant Hadamard matrices Barker sequences Field descent 

Mathematics Subject Classification (2000)

15B34 05B10 


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  1. 1.
    Beth T, Jungnickel D, Lenz H.: Design Theory, 2nd edn. Cambridge University Press, Cambridge (1999)CrossRefGoogle Scholar
  2. 2.
    Borevich Z.I, Shafarevich I.R.: Number Theory. Academic Press, New York (1966)Google Scholar
  3. 3.
    Chan W.K.: Necessary conditions for menon difference sets. Des. Codes Cryptogr 3, 147–154 (1993)MathSciNetMATHCrossRefGoogle Scholar
  4. 4.
    Eliahou S, Kervaire M, Saffari B.: A new restriction on the length of Golay complementary sequences. J. Comb. Theory Ser A 55, 49–59 (1990)MathSciNetMATHCrossRefGoogle Scholar
  5. 5.
    Leung K.H, Schmidt B.: The field descent method. Des. Codes Cryptogr 36, 171–188 (2005)MathSciNetMATHCrossRefGoogle Scholar
  6. 6.
    McFarland R.L.: Sub-difference sets of Hadamard difference sets. J. Comb. Theory A 54, 112–122 (1990)MathSciNetMATHCrossRefGoogle Scholar
  7. 7.
    Mossinghoff M.: Wieferich pairs and Barker sequences. Des. Codes Cryptogr 53, 149–163 (2009)MathSciNetMATHCrossRefGoogle Scholar
  8. 8.
    Ryser H.J.: Combinatorial Mathematics. Wiley, New York (1963)MATHGoogle Scholar
  9. 9.
    Schmidt B.: Cyclotomic integers and finite geometry. J. Am. Math. Soc 12, 929–952 (1999)MATHCrossRefGoogle Scholar
  10. 10.
    Schmidt B.: Characters and cyclotomic fields in finite geometry. Lecture Notes in Mathematics, vol. 1797. Springer, Berlin (2002)CrossRefGoogle Scholar
  11. 11.
    Schmidt B.: Data on Circulant Hadamard Matrices.
  12. 12.
    Storer J, Turyn R.: On binary sequences. Proc. Am. Math. Soc 12, 394–399 (1961)MathSciNetMATHCrossRefGoogle Scholar
  13. 13.
    Turyn R.J.: Character sums and difference sets. Pacific J. Math 15, 319–346 (1965)MathSciNetMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2011

Authors and Affiliations

  1. 1.Department of MathematicsNational University of SingaporeSingaporeRepublic of Singapore
  2. 2.Division of Mathematical Sciences, School of Physical & Mathematical SciencesNanyang Technological UniversitySingaporeRepublic of Singapore

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