\mathbb{F}_{p^{n}} \}$ is called graph of $f$. Any non-principal character $\chi$ of the additive group of $\mathbb{F}_{p^{n}}\times \mathbb{F}_p$ satisfies $|\chi(G_f)|^2=p^n$ or $0$. This observation yields that $G_f$ is a $(p^{n+1},p^n, p^{2n-1}-p^{n-1},p^{2n-1}-p^{n-1}+p^n)$-PGDS in $H=\mathbb{F}_{p^{n}}\times\mathbb{F}_{p}$.

Walsh transform provides interesting connections between $p$-ary functions and difference sets. For a prime $p$, we define a primitive complex $p$-th root of unity $\zeta_p=e^{\frac{2\pi i}{p}}$. Let $f$ be a function from the field $\mathbb{F}_{p^{n}}$ to $\mathbb{F}_p$ and let $F(x)=\zeta_p^{f(x)}$. The Walsh transform of $f$ is defined as follows: $$\displaystyle \widehat{f}(\mu)=\sum_{x\in \mathbb{F}_{p^{n}}}  \zeta_p^{f(x)-Tr_n(\mu x)},~~~\mu \in \mathbb{F}_{p^{n}}$$ where $$Tr_n(z)=\sum_{i=0}^{n-1}z^{p^i}.$$

The convolution of $F$ and $G$ is defined by $$(F * G)(a)=\sum_{x\in \mathbb{F}_{p^n} } F(x)G(x-a).$$ We will also take advantage of the convolution theorem of Fourier analysis. The Fourier transform of a convolution of two functions is $$\widehat{F * G}=\widehat{F}\cdot \widehat{G}.$$

Results on Vectorial Functions
==============================

Let $F$ be a vectorial function from $\mathbb{F}_{p^{n}}$ to $\mathbb{F}_{p^{m}}$. For every $b\in \mathbb{F}^*_{p^{m}}$, the component function $F_b$ of $F$ from $\mathbb{F}_{p^{n}}$ to $\mathbb{F}_p$ is defined as $F_{b}(x)=Tr_m(bF(x))$. A vectorial function is called *vectorial plateaued* if all its nonzero component functions are plateaued. If the nonzero component functions of a vectorial plateaued function are s-plateaued for the same $0 \leq s \leq n$ 