between the parameters of a PGDS and the second order derivatives of (vectorial) plateaued functions.

A family of vectorial $s-$plateaued functions
---------------------------------------------

In this section, we will discuss the link between vectorial $s$-plateaued functions $F(x)=x^d$ from $\mathbb{F}_{p^{n}}$ to $\mathbb{F}_{p^{n}}$ and the cross-correlation function between two $p$-ary $m$-sequences that differ by a decimation $d$. An $m$-sequence and its decimation is defined by $u(t)=\sigma^t$ and $v(t)=u(dt)$ where $\sigma$ is a primitive element of the finite field. The cross-correlation between the sequences $u$ and $v$ is defined by $$\theta(\tau)=\sum_{t=0}^{p^n-2}\zeta_p^{u(t+\tau)-v(t)}.$$ It can be shown that $$\theta(\tau)=-1+\sum_{x \in \mathbb{F}_{p^{n}} }\zeta_p^{Tr_n(ax+x^d)}$$ where $a=-\sigma^\tau$. Therefore for $F_1(x)=Tr(x^d)$, we have $$\label{CC-Walsh}
 \theta(\tau)=-1+\widehat{F_1}(-a).$$

\[cross-plateaued\] Let $F(x)=x^{d}$ be a vectorial function from $\mathbb{F}_{p^{n}}$ to $\mathbb{F}_{p^{n}}$ with $gcd(d,p^n-1)=1$. If the cross-correlation of the $p$-ary $m$-sequences that differ by decimation $d$ takes three values, namely $-1, -1+p^{\frac{n+s}{2}}$ and $-1-p^{\frac{n+s}{2}}$, then $F$ is a vectorial $s$-plateaued function.

For each $b\in \mathbb{F}_{p^{n}}^*$, we denote by $F_b(x)$ the function $F_b(x)=Tr(bF(x))=Tr(bx^d)$. The walsh transform $$\widehat{F_b}(0)=\sum_{x \in \mathbb{F}_{p^{n}}} \zeta_p^{Tr_n(bx^d)}=0$$ since $x^d$ is a permutation. For each $a\in \mathbb{F}_{p^{n}}^*$, the Walsh transform of $F_b(x)$ $$\begin{aligned}
    \widehat{F_b}(a)&=\sum_{x \in \mathbb{F}_{p^{n}}} \zeta_p^{Tr_n(bx^d-ax)}\\
    &=\sum_{x \in\mathbb{F}_{p^{n}}  }\zeta_p^{Tr_n(c^dx^d-\frac{a}{c}cx)}\\
    &=\sum_{y \in\mathbb{F}_{p^{n}}  }\zeta_p^{Tr_n(y^d-\mu y)}\\
    &=\widehat{F_1}(\mu)
    \end{aligned}$$ where $b=c^d$ and $\mu=a/c$. Note that any $b\in \mathbb{F}_{p^{n}}^*$ can 