Geodesic
Criteria for maximal nonlinearity of a function over a finite field
Diskretnaya Matematika, Tome 33 (2021) no. 3, pp. 79-91

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An $n$-place function over a field with $q$ elements is called maximally nonlinear if it has the greatest nonlinearity among all such functions. Criteria and necessary conditions for maximal nonlinearity are obtained, which imply that, for even $n$, the maximally nonlinear functions are bent functions, but, for $q>2$, the known families of bent functions are not maximally nonlinear. For an arbitrary finite field, a relationship between the Hamming distances from a function to all affine mappings and the Fourier spectra of the nontrivial characters of the function are found.
Keywords: finite field, nonlinearity, affine function, bent function, Fourier coefficients. 
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     author = {V. G. Ryabov},
     title = {Criteria for maximal nonlinearity of a function over a finite field},
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V. G. Ryabov. Criteria for maximal nonlinearity of a function over a finite field. Diskretnaya Matematika, Tome 33 (2021) no. 3, pp. 79-91. http://geodesic.mathdoc.fr/item/DM_2021_33_3_a5/