On q-ary bent and plateaued functions

Abstract

We obtain the following results. For any prime p the minimal Hamming distance between distinct regular p-ary bent functions of 2n variables is equal to $p^n$. The number of p-ary regular bent functions at the distance $p^n$ from the quadratic bent function $Q_n=x_1x_2+\cdots +x_{2n-1}x_{2n}$ is equal to $p^n(p^{n-1}+1)\cdots (p+1)(p-1)$ for $p>2$. The Hamming distance between distinct binary s-plateaued functions of n variables is not less than $2^{\frac{s+n-2}{2}}$ and the Hamming distance between distinct ternary s-plateaued functions of n variables is not less than $3^{\frac{s+n-1}{2}}$. These bounds are tight. For $p=3$ we prove an upper bound on nonlinearity of ternary functions in terms of their correlation immunity. Moreover, functions reaching this bound are plateaued. For $p=2$ analogous result are well known but for large p it seems impossible. Constructions and some properties of p-ary plateaued functions are discussed.