Approximation of plateaued boolean functions by monomial ones

From the Parseval's equation we have that if the squared Walsh transform of the Boolean function takes at most one nonzero value then its Walsh coefficients are equal to $2^{2n-2s}$ for some $s\le n/2$. These functions are called the $2s$-order plateaued functions. In the present paper we consider the aspects of an approximation of the plateaued functions by monomial ones. We use the representation of $n$-variable Boolean functions by polynomials over the field $\mathbb F_{2^n}$. The necessary conditions for the Boolean functions to have the Hamming distance to all bijective monomials taking only three values: $2^{n-1}$, $2^{n-1}\pm 2^{n-s-1}$, are obtained. The non-existence of the functions, satisfying these conditions for such $n$ that $2^n-1$ is prime, is shown.