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Bent functions from a vector space $${{\mathbb {V}}}_n$$ V n over $${{\mathbb {F}}}_2$$ F 2 of even dimension $$n=2m$$ n = 2 m into the cyclic group $${{\mathbb {Z}}}_{2^k}$$ Z 2 k , or equivalently, relative difference sets in $${{\mathbb {V}}}_n\times {{\mathbb {Z}}}_{2^k}$$ V n × Z 2 k with forbidden subgroup $${{\mathbb {Z}}}_{2^k}$$ Z 2 k , can be obtained from spreads of $${{\mathbb {V}}}_n$$ V n for any $$k\le n/2$$ k ≤ n / 2 . In this article, existence and construction of bent functions from $${{\mathbb {V}}}_n$$ V n to $${{\mathbb {Z}}}_{2^k}$$ Z 2 k , which do not come from the spread construction is investigated. A construction of bent functions from $${{\mathbb {V}}}_n$$ V n into $${{\mathbb {Z}}}_{2^k}$$ Z 2 k , $$k\le n/6$$ k ≤ n / 6 , (and more generally, into any abelian group of order $$2^k$$ 2 k ) is obtained from partitions of $${{\mathbb {F}}}_{2^m}\times {{\mathbb {F}}}_{2^m}$$ F 2 m × 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.

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