be written as $b=c^d$ for some $c \in \mathbb{F}_{p^{n}}^*$. Then using Equation (\[CC-Walsh\]), we immediately obtain the result that $F(x)$ is vectorial s-plateaued.

Let $p$ be an odd prime and $n,k$ be positive integers with $gcd(n,k)=s$. If $ n/s$ is odd then $\gcd{(p^n-1,d)}=1$ for $d=(p^{2k}+1)/2$ and $d=p^{2k}-p^k+1$.

There are only finitely many known functions with a three-valued cross-correlation. Trachtenberg proved the following in his thesis [@trachtenberg]. Let $n$ be an odd integer and $k$ be an integer such that $gcd(n,k)=s$. Then for each of the decimations $d=\frac{p^{2k}+1}{2}$ and $d=p^{2k}-p^k+1$ the cross-correlation function $\theta_d(\tau)$ takes the values $-1,-1\pm p^{\frac{n+s}{2}}$.

This result is generalized later by Helleseth in Theorem 4.9 in [@helleseth]. He showed that if $gcd(n,k)=s$ and $ n/s$ is odd then for the same decimations, $\theta_d(\tau)$ has the values $-1,-1\pm p^{\frac{n+s}{2}}$. Our result implies that the corresponding vectorial functions are $s-$plateaued.

Pott et. al. provided a classification of weakly regular bent functions via partial difference sets, [@tan2010]. In this classification authors showed that a function from $\mathbb{F}_3^n$ to $\mathbb{F}_3$ when $n$ is even satisfying $f(-x) = f(x)$ and $f(0)=0$ is weakly regular if and only if a $D_1=\{x: f(x)=1\}$ and $D_2=\{x: f(x)=2\}$ are partial difference sets. Later in [@olmez3], the 