is well known that bent functions exist if and only if $n$ is even. To find bent functions is a hard task. So far the known largest class of bent functions is the so called Maiorana-McFarland class of bent functions (denoted by $\mathcal{M}$) [@McF], which are of the form $f(x,y)={\rm Tr}^{k}_{1}(\pi(y)x)+g(y)$, $(x,y)\in \mathbb{F}_{2^{k}}\times  \mathbb{F}_{2^{k}}$, where $k$ is some positive integer, $\pi$ is a permutation over $\mathbb{F}_{2^{k}}$ and $g$ is an arbitrary Boolean function in $k$ variables. The function $f$ is called [*plateaued*]{} when either it is bent or $W_f$ takes three values $\{0,\pm 2^{s}\}$ for some integer $n/2< s\leq n$.

The nonlinearity of an $(n,m)$-function $F$ and hereby its resistance to [*linear cryptanalysis*]{} [@Matsui] is measured through the [*extended Walsh spectrum*]{} $$\{\ast~ |W_{F}(\sigma,\gamma)|:\gamma \in \mathbb{F}^{m}_{2}\backslash\{0\},\sigma \in \mathbb{F}^{n}_{2} ~\ast\},$$ where $$W_{F}(\sigma,\gamma)\!=\!\sum\limits_{x\in  \mathbb{F}^{n}_{2}}(\!-\!1)^{\langle \sigma,x\rangle\! +\!\langle \gamma,F(x)\rangle }.$$ The function $F$ is said to be a [*vectorial bent function*]{} of dimension $m$ if all the components of $F$ are bent. In other words, $F$ is vectorial bent if and only if $|W_{F}(\sigma,\gamma)|=2^{n/2}$, for any $\gamma \in \mathbb{F}^{m}_{2}\backslash\{0\}$ and for any $\sigma\in\mathbb{F}^{n}_{2}$. $F$ is said to be a [*plateaued vectorial function*]{} if all the components are plateaued Boolean functions.

For an $(n,m)$-function $F$, denote 