On q-ary bent and plateaued functions

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.