On the question on the approximation of vectorial functions over finite fields by affine analogues

The measure of closeness of vectorial functions is defined by the Hamming distance in the space of their values, and the nonlinearity of a vector function is defined as the Hamming distance to the set of affine mappings. Bounds and estimates for the distribution of nonlinearity of balanced mappings and substitutions are obtained. Classes of vector functions with high nonlinearity are constructed. The nonlinearity introduced in this way is compared with the nonlinearity defined as the minimal nonlinearity over all nontrivial linear combinations of coordinate functions.