We study vectorial functions with maximal number of bent components in this paper. We first study the Walsh transform and nonlinearity of \(F(x)=x^{2^e}h(\textrm{Tr}_{2^{2m}/2^m}(x))\) , where \(e\ge 0\) and h(x) is a permutation over \({\mathbb {F}}_{2^m}\) . If h(x) is monomial, the nonlinearity of F(x) is shown to be at most \( 2^{2\,m-1}-2^{\lfloor \frac{3\,m}{2}\rfloor }\) and some non-plateaued and plateaued functions attaining the upper bound are found. 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 quadratic vectorial function that is not equivalent to the known functions of the form xh(x), and also contains vectorial functions outside the completed Maiorana-McFarland class. Finally, we show that the vectorial function \(F: {\mathbb {F}}_{2^{2m}}\rightarrow {\mathbb {F}}_{2^{2m}}\) , \(x\mapsto x^{2^m+1}+x^{2^i+1}\) has maximal number of bent components if and only if \(i=0\) .
