On functions with the maximal number of bent components

A function $F:\mathbb{F}2^n\rightarrow \mathbb{F}2^n$, $n=2m$, can have at most $2^n-2m$ bent component functions. Trivial examples are obtained as $F(x) = (f1(x),\ldots,fm(x),a1(x),\ldots, am(x))$, where $\tilde{F}(x)=(f1(x),\ldots,fm(x))$ is a vectorial bent function from $\mathbb{F}2^n$ to $\mathbb{F}2^m$, and $ai$, $1\le i\le m$, are affine Boolean functions. A class of nontrivial examples is given in univariate form with the functions $F(x) = x^{2r}{\rm Trⁿm}(\Lambda(x))$, where $\Lambda$ is a linearized permutation of $\mathbb{F}{2^m}$. In the first part of this article it is shown that plateaued functions with $2^n-2m$ bent components can have nonlinearity at most $2^{{n-1}-2{\lfloor\frac{n+m}{2}\rfloor}$,} a bound which is attained by the example $x^{2r}{\rm Trⁿm}(x)$, $1\le r<m$ (Pott et al. 2018). This partially solves Question 5 in Pott et al. 2018. We then analyse the functions of the form $x^{2r}{\rm Tr^{n_m}(\Lambda(x))$.} We show that for odd $m$, only $x^{2r}{\rm Tr^n_m}(x)$, $1\le r<m$, has maximal nonlinearity, whereas there are more of them for even $m$, of which we present one more infinite class explicitly. In detail, we investigate Walsh spectrum, differential spectrum and their relations for the functions $x^{2r}{\rm Tr^{n_m}(\Lambda(x))$.} Our results indicate that this class contains many nontrivial EA-equivalence classes of functions with the maximal number of bent components, if $m$ is even, several with maximal possible nonlinearity.