Criteria for maximal nonlinearity of a function over a finite field

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.