Graphs of vectorial plateaued functions as difference sets

A function F:Fpn→Fpm, is a vectorial s-plateaued function if for each component function Fb(μ)=Trn(bF(x)),b∈Fpm⁎ and μ∈Fpn, the Walsh transform value |Fbˆ(μ)| is either 0 or pn+s2. In this paper, we explore the relation between (vectorial) s-plateaued functions and partial geometric difference sets. Moreover, we establish the link between three-valued cross-correlation of p-ary sequences and vectorial s-plateaued functions. Using this link, we provide a partition of F3n into partial geometric difference sets. Conversely, using a partition of F3n into partial geometric difference sets, we construct ternary plateaued functions f:F3n→F3. We also give a characterization of p-ary plateaued functions in terms of special matrices which enables us to give the link between such functions and second-order derivatives using a different approach.