Vectorial Bent-Negabent Functions—Their Constructions and Bounds

Boolean bent functions which at the same time have a flat nega-Hadamard transform are called bent-negabent functions. The known families of these functions mostly stem from the Maiorana-McFarland class of bent functions and their vectorial counterparts have not been considered in the literature. In this article, we introduce the notion of
vectorial bent-negabent
functions and show that in general for a vectorial bent-negabent function
$F\colon {\mathbb {F}} _{2}^{2m} \rightarrow {\mathbb {F}} _{2}^{k}$
we necessarily have that
$k \leq m-1$
. We specify a class of vectorial bent-negabent functions of maximal output dimension
$m-1$
by using a set of linear complete mappings. On the other hand, we propose several methods (one of which is generic) of specifying vector spaces of nonlinear complete mappings which then induce vectorial bent-negabent functions (whose dimension is not maximal) having a certain number of component functions outside the completed Maiorana-McFarland class. Finally, we derive an upper bound on the maximum number of bent-negabent components for mappings
$F\colon {\mathbb {F}} _{2}^{2m} \rightarrow {\mathbb {F}} _{2}^{k}$
, where
$m \leq k \leq 2m$
, and identify some families of these functions reaching this upper bound.