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

Abstract

Bent functions from a vector space ${{\mathbb {V}}}_n$ over ${{\mathbb {F}}}_2$ of even dimension $n=2m$ into the cyclic group ${{\mathbb {Z}}}_{2^k}$, or equivalently, relative difference sets in ${{\mathbb {V}}}_n\times {{\mathbb {Z}}}_{2^k}$ with forbidden subgroup ${{\mathbb {Z}}}_{2^k}$, can be obtained from spreads of ${{\mathbb {V}}}_n$ for any $k\le n/2$. In this article, existence and construction of bent functions from ${{\mathbb {V}}}_n$ to ${{\mathbb {Z}}}_{2^k}$, which do not come from the spread construction is investigated. A construction of bent functions from ${{\mathbb {V}}}_n$ 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.