Bent and ℤ_2^k-bent functions from spread-like partitions
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Bent functions from a vector space V_n over 𝔽_2 of even dimension n=2m into the cyclic group ℤ_2^k, or equivalently, relative difference sets in V_n×ℤ_2^k with forbidden subgroup ℤ_2^k, can be obtained from spreads of V_n for any k≤ n/2. In this article, existence and construction of bent functions from V_n to ℤ_2^k, which do not come from the spread construction is investigated. A construction of bent functions from V_n into ℤ_2^k, k≤ n/6, (and more generally, into any abelian group of order 2^k) is obtained from partitions of 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.
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