On vectorial functions with maximal number of bent components

We study vectorial functions with maximal number of bent components in this paper. We first study the Walsh transform and nonlinearity of $F(x)=x^{2^e}h({\text{Tr}}_{2^{2m}\mathrm{/}{2}^m}(x))$, where $e\ge 0$ and $h(x)$ is a permutation over ${\text{F}}_{2^m}$. If $h(x)$ is monomial, the nonlinearity of $F(x)$ is shown to be at most $2^{2m-1}-2^{\lfloor \frac{3m}{2}\rfloor }$ and some non-plateaued and plateaued functions attaining the upper bound are found. This gives a partial answer to the open problems proposed by Pott et al. and Anbar et al. If $h(x)$ is linear, the exact nonlinearity of $F(x)$ is determined. Secondly, we give a construction of vectorial functions with maximal number of bent components from known ones, thus obtain two new classes from the Niho class and the Maiorana-McFarland class. Our construction gives a partial answer to an open problem proposed by Pott et al., and also contains vectorial functions outside the complete Maiorana-McFarland class. Finally, we show that the vectorial function $F:{\text{F}}_{2^{2m}}\to {\text{F}}_{2^{2m}}$, $x\mapsto x^{2^m+1}+x^{2^i+1}$ has maximal number of bent components if and only if $i=0$.