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.