ИСТИНА

A function of n variables over a finite field of q elements is called maximally nonlinear if it has the greatest nonlinearity among all q-valued functions of n variables. It is proved that for q > 2 and even values n, a necessary condition for the maximum nonlinearity of a function is the absence of a linear manifold of dimension greater than or equal to n/2, on which its restriction would coincide with the restriction of some affine function. In accordance with it, for q > 2, functions from Maiorana-McFarland's and Dillon's families of bent functions are not maximally nonlinear. A new family of maximally nonlinear bent functions of degree from 2 to max {2, (q-1)(n/2-1)} with nonlinearity equal (q-1)q^{n-1} - q^{n/2-1} is constructed.