Designs, Codes and Cryptography

, Volume 74, Issue 3, pp 665–672 | Cite as

Highly nonlinear functions

  • Kai-Uwe SchmidtEmail author


Let f be a function from \(\mathbb {Z}_q^m\) to \(\mathbb {Z}_q\). Such a function f is bent if all values of its Fourier transform have absolute value 1. Bent functions are known to exist for all pairs \((m,q)\) except when m is odd and \(q\equiv 2\pmod 4\) and there is overwhelming evidence that no bent function exists in the latter case. In this paper the following problem is studied: how closely can the largest absolute value of the Fourier transform of f approach 1? For \(q=2\), this problem is equivalent to the old and difficult open problem of determining the covering radius of the first order Reed–Muller code. The main result is, loosely speaking, that the largest absolute value of the Fourier transform of f can be made arbitrarily close to 1 for q large enough.


Generalised bent function Nonlinearity Fourier coefficient  Probabilistic method 

Mathematics Subject Classification

06E75 42A16 05D40 


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Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  1. 1.Faculty of MathematicsOtto-von-Guericke UniversityMagdeburgGermany

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