$x,y \in \mathbb{F}_{p^{n+m}}$ then $x=\gamma_i+u$ and $y=\gamma_j+v$ for unique elements $u,v \in H$. Now let us examine the $xy$-th entry of $P$. $$P_{x,y}=\zeta_p^{f(\phi_2(\gamma_i+H)+\phi_2(\gamma_j+H))} \cdot \zeta_p^{g(\phi_1(u)+\phi_1(v))}= \zeta_p^{h(x+y)}$$ where $h$ is the desired $(s_1+s_2)$-plateaued function from $\mathbb{F}_{p^{n+m}}$ to $\mathbb{F}_{p}.$ Moreover if $y=0$ then $\gamma_j=v=0$. Thus $$h(x)= f(\phi_2(\gamma_i+H))+g(\phi_1(u)).$$

Partially bent functions 
-------------------------

This section is devoted to investigate a family of $s-$plateaued functions known as partially bent functions. A $p-$ary function is called partially bent if the derivative $D_af$ is either balanced or constant for any $a \in \mathbb{F}_{p^n}$. Here we will provide some characterization via their associated designs.

Let $f$ be an $s$-plateaued function and for $a \in \mathbb{F}_{p^{n}} $ define the set $$T_a=\{x+a,f(x)+f(a): x \in \mathbb{F}_{p^{n}} \}.$$ Fix an element $a$ of $\mathbb{F}_{p^{n}} $, then the graph of $f$ can be also written as $$G_f=\{(x,f(x)): x \in \mathbb{F}_{p^{n}}\}=\{(x+a,f(x+a)): x \in \mathbb{F}_{p^{n}}\}.$$

Let $f$ be an $s$-plateaued function with $f(0)=0$. If $f$ has a linear structure $\Lambda$ then $T_a=G_f$ for all $a \in \Lambda$.

Let $f$ be an $s$-plateaued function with $f(0)=0$ and linear structure $\Lambda$ of dimension $m$. Then the incidence matrix $A$ of the design associated with the partial geometric difference set $G_f$ can be written as a Kronecker product 