Approximation of vectorial functions over finite fields and their restrictions to linear manifolds by affine analogues

The nonlinearity of vectorial functions and of their restrictions to manifolds are defined as the Hamming distance to the set of affine mappings and of their restrictions to the manifold, respectively. Relations between the parameters of the nonlinearity of a vectorial function and their analogues for its coordinate functions and its restrictions to manifolds are established. An analogue of the Parseval identity for such parameters of vectorial functions is proved, which implies the upper bound $(q^k-1)q^{n-k}-q^{n/2-k}$ for the nonlinearity of a mapping over a $q$-element field of $n$ variables with $k$ coordinates. Attainability conditions for this estimate are found, and a class of Boolean vectorial functions with high value of nonlinearity is constructed. Estimates characterizing the distribution of the nonlinearity of a vectorial function and of its restrictions to manifolds are obtained.