Cryptography and Communications

, Volume 2, Issue 2, pp 307–334 | Cite as

A Fourier-analytic approach to counting partial Hadamard matrices



In this paper, we study a family of lattice walks which are related to the Hadamard conjecture. There is a bijection between paths of these walks which originate and terminate at the origin and equivalence classes of partial Hadamard matrices. Therefore, the existence of partial Hadamard matrices can be proved by showing that there is positive probability of a random walk returning to the origin after a specified number of steps. Moreover, the number of these designs can be approximated by estimating the return probabilities. We use the inversion formula for the Fourier transform of the random walk to provide such estimates. We also include here an upper bound, derived by elementary methods, on the number of partial Hadamard matrices.


Partial Hadamard matrices Random walks 

Mathematics Subjects Classifications (2010)

05B20 15B10 15B34 60G50 


  1. 1.
    Billingsley, P.: Probability and measure. In: Wiley Series in Probability and Mathematical Statistics, 3rd edn. Wiley, New York. A Wiley-Interscience Publication (1995)Google Scholar
  2. 2.
    Canfield, E.R., Gao, Z., Greenhill, C., McKay, B.D., Robinson, R.W.: Asymptotic enumeration of correlation-immune Boolean functions. Cryptogr. Commun. 2(1), 111–126 (2010)MATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Canfield, E.R., McKay, B.D.: Asymptotic enumeration of dense 0–1 matrices with equal row sums and equal column sums. Electron. J. Comb. 12, Research paper 29, 31 (2005)Google Scholar
  4. 4.
    Craigen, R.: Signed groups, sequences, and the asymptotic existence of Hadamard matrices. J. Comb. Theory, Ser. A. 71(2), 241–254 (1995)MATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    de Launey, W., Gordon, D.M.: A comment on the Hadamard conjecture. J. Comb. Theory, Ser. A. 95(1), 180–184 (2001)MATHCrossRefGoogle Scholar
  6. 6.
    Graham, S.W., Shparlinski, I.E.: On RSA moduli with almost half of the bits prescribed. Discrete Appl. Math. 156(16), 3150–3154 (2008)MATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    McKay, B.D., Wormald, N.C.: Asymptotic enumeration by degree sequence of graphs of high degree. Eur. J. Comb. 11(6), 565–580 (1990)MATHMathSciNetGoogle Scholar
  8. 8.
    Spitzer, F.: Principles of random walks. In: Graduate Texts in Mathematics, vol. 34, 2nd edn. Springer, New York (1976)Google Scholar

Copyright information

© Springer Science + Business Media, LLC 2010

Authors and Affiliations

  1. 1.Center for Communication ResearchSan DiegoUSA
  2. 2.Department of MathematicsUniversity of OregonEugeneUSA

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