CHARACTERIZATION OF CERTAIN TYPES OF r-PLATEAUED FUNCTIONS

We study a subclass of p-ary functions in n variables, denoted by ${\mathcal{A}}_n$, which is a collection of p-ary functions in n variables satisfying a certain condition on the exponents of its monomial terms. Firstly, we completely classify all p-ary (n - 1)-plateaued functions in n variables by proving that every (n - 1)-plateaued function should be contained in ${\mathcal{A}}_n$. Secondly, we prove that if f is a p-ary r-plateaued function contained in ${\mathcal{A}}_n$ with deg f > $1+{\frac{n-r}{4}}(p-1)$, then the highest degree term of f is only a single term. Furthermore, we prove that there is no p-ary r-plateaued function in ${\mathcal{A}}_n$ with maximum degree $(p-1){\frac{n-4}{2}}+1$. As application, we partially classify all (n - 2)-plateaued functions in ${\mathcal{A}}_n$ when p = 3, 5, and 7, and p-ary bent functions in ${\mathcal{A}}_2$ are completely classified for the cases p = 3 and 5.