On number of substitutions of vector space over finite field with affine approximations with given accuracy

Nonlinear mapping of a vector space over a finite field in vector space over the same field and, in particular, the non-linearity the permutation space is defined in terms of the Hamming distance to sets of affine mappings. For permutations of an n-dimensional vector space over a field of q elements, an upper bound for the number permutations with non-linearity, not higher than a given value r, where 0 ≤ r < q<sup>n</sup> - q<sup>n-1</sup>, and for r < (q<sup>n</sup> - q<sup>n-1</sup>)/2, the exact the value of the specified number, which allows us to draw conclusions about distribution of nonlinearity on the set of substitutions.