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New restrictions on possible orders of circulant Hadamard matrices

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Abstract

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.

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References

  1. Beth T, Jungnickel D, Lenz H.: Design Theory, 2nd edn. Cambridge University Press, Cambridge (1999)

    Book  Google Scholar 

  2. Borevich Z.I, Shafarevich I.R.: Number Theory. Academic Press, New York (1966)

    Google Scholar 

  3. Chan W.K.: Necessary conditions for menon difference sets. Des. Codes Cryptogr 3, 147–154 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  5. Leung K.H, Schmidt B.: The field descent method. Des. Codes Cryptogr 36, 171–188 (2005)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  7. Mossinghoff M.: Wieferich pairs and Barker sequences. Des. Codes Cryptogr 53, 149–163 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  8. Ryser H.J.: Combinatorial Mathematics. Wiley, New York (1963)

    MATH  Google Scholar 

  9. Schmidt B.: Cyclotomic integers and finite geometry. J. Am. Math. Soc 12, 929–952 (1999)

    Article  MATH  Google Scholar 

  10. Schmidt B.: Characters and cyclotomic fields in finite geometry. Lecture Notes in Mathematics, vol. 1797. Springer, Berlin (2002)

    Book  Google Scholar 

  11. Schmidt B.: Data on Circulant Hadamard Matrices. http://www3.ntu.edu.sg/home/Bernhard/CW/CW.html.

  12. Storer J, Turyn R.: On binary sequences. Proc. Am. Math. Soc 12, 394–399 (1961)

    Article  MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

Download references

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Authors and Affiliations

  1. Department of Mathematics, National University of Singapore, Kent Ridge, Singapore, 119260, Republic of Singapore

    Ka Hin Leung

  2. Division of Mathematical Sciences, School of Physical & Mathematical Sciences, Nanyang Technological University, Singapore, 637371, Republic of Singapore

    Bernhard Schmidt

Authors
  1. Ka Hin Leung
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  2. Bernhard Schmidt
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Corresponding author

Correspondence to Bernhard Schmidt.

Additional information

This is one of several papers published together in Designs, Codes and Cryptography on the special topic: “Geometry, Combinatorial Designs & Cryptology”.

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Leung, K.H., Schmidt, B. New restrictions on possible orders of circulant Hadamard matrices. Des. Codes Cryptogr. 64, 143–151 (2012). https://doi.org/10.1007/s10623-011-9493-1

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  • DOI: https://doi.org/10.1007/s10623-011-9493-1

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