Bent and $\mathbb Z_{2^k}$-bent functions from spread-like partitions

Bent functions from a vector space $Vn$ over $\mathbb F2$ of even dimension $n=2m$ into the cyclic group $\mathbb Z{2^k}$, or equivalently, relative difference sets in $Vn\times\mathbb Z{2^k}$ with forbidden subgroup $\mathbb Z{2^k}$, can be obtained from spreads of $Vn$ for any $k\le n/2$. In this article, existence and construction of bent functions from $Vn$ to $\mathbb Z{2^k}$, which do not come from the spread construction is investigated. A construction of bent functions from $Vn$ into $\mathbb Z{2^k}$, $k\le n/6$, (and more generally, into any abelian group of order $2^k$) is obtained from partitions of $\mathbb F{2^{m}\times\mathbb} F_{2^m}$, which can be seen as a generalization of the Desarguesian spread. As for the spreads, the union of a certain fixed number of sets of these partitions is always the support of a Boolean bent function.