Bent and Z_(2k)-Bent functions from spread-like partitions

摘要

Bent functions from a vector space V-n over F-2 of even dimension n = 2m into the cyclic group Z(2k), or equivalently, relative difference sets in V-n xZ(2k) with forbidden subgroup Z2k, can be obtained from spreads ofV(n) for any k = n/2. In this article, existence and construction of bent functions from V-n to Z(2k), which do not come from the spread construction is investigated. A construction of bent functions from Vn into Z(2k), k = n/6, (and more generally, into any abelian group of order 2(k)) is obtained from partitions of F-2m x F-2m, 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.