[@Mes16].

The bent property of Boolean functions has been extended to general $(n,m)$-functions by requesting that all the nonzero linear combinations of the [*coordinate functions*]{} of $F$ are bent functions. Such functions are called [*vectorial bent*]{}. The construction of vectorial bent functions has been initially considered by Nyberg [@Nyb]. It was shown in [@Nyb] that vectorial bent $(n,m)$-functions exist if and only if $n$ is even and $m\leq n/2$. Such functions can be constructed by using some known classes of bent functions, such as the [*Maiorana-McFarland class*]{} (denoted by $\mathcal{M}$) and the Dillon’s [*partial spread class*]{} (denoted by $\mathcal{PS}$). In the finite field version, the bent property of $(n,m)$-functions $F: \mathbb{F}_{2^{n}} \rightarrow \mathbb{F}_{2^{m}}$ with $\mathbb{F}_{2^{m}}\subseteq \mathbb{F}_{2^{n}}$ can be established directly as it was done in [@MPS] for the trace functions ${\rm Tr}^{n}_{k}(\sum\limits^{2^{k}}_{i=0}\alpha_{i}x^{i(2^{k}-1)})$, where $n=2k$ and $\alpha_{i}\in \mathbb{F}_{2^{n}}$, see [@Mes16] for more constructions of vectorial bent functions.

An interesting and important problem concerning bent functions and vectorial bent functions is that, how large could the number of [*bent components*]{} of an $(n,m)$-function be ($n$ being even)? Thanks to Nyberg’s result, the largest value of this number is $2^m-1$ when $m\leq n/2$. While for $m>n/2$, the answer is unknown before. Very recently, 