On design-theoretic aspects of Boolean and vectorial bent functions

There are two construction methods of designs from $(n,m)$-bent functions, known as translation and addition designs. In this paper we analyze, which equivalence relation for Boolean bent functions, i.e. $(n,1)$-bent functions, and vectorial bent functions, i.e. $(n,m)$-bent functions with $2\le m\le n/2$, is coarser: extended-affine equivalence or isomorphism of associated translation and addition designs. First, we observe that similar to the Boolean bent functions, extended-affine equivalence of vectorial $(n,m)$-bent functions and isomorphism of addition designs are the same concepts for all even $n$ and $m\le n/2$. Further, we show that extended-affine inequivalent Boolean bent functions in $n$ variables, whose translation designs are isomorphic, exist for all $n\ge6$. This implies, that isomorphism of translation designs for Boolean bent functions is a coarser equivalence relation than extended-affine equivalence. However, we do not observe the same phenomenon for vectorial bent functions in a small number of variables. We classify and enumerate all vectorial bent functions in six variables and show, that in contrast to the Boolean case, one cannot exhibit isomorphic translation designs from extended-affine inequivalent vectorial $(6,m)$-bent functions with $m\in\{ 2,3 \}$.