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

Abstract

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

20 pages