On the invertibility of vector Boolean functions

The class $\mathcal F_{n,m,k}$ of invertible vector Boolean functions $F\colon\mathbb F_2^n\to\mathbb F_2^m$ with coordinate functions depending on the given number $k$ variables is considered. It is proved that 1) these functions do not exist for any $n=m$ and $k=2$; 2) the functions of the class $\mathcal F_{n,n,n-1}$ can (can not) be built from affine coordinate functions for even (odd) $n$; 3) if $\mathcal F_{n,m,k}\neq\varnothing$ then $\mathcal F_{n+1,m+1,k}\neq\varnothing$.