Bent and \mathbb Z_2^k-bent functions from spread-like partitions

2020/09/23 by Meidl, Wilfried, Pirsic, Isabel
#05B10 #06E30 #94C10 #FOS: Computer and information sciences #FOS: Mathematics #Information Theory (cs.IT) #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2009.11019

Abstract

Bent functions from a vector space V_n over \mathbb F₂ of even dimension n=2m into the cyclic group \mathbb Z_2^k, or equivalently, relative difference sets in V_n×\mathbb Z_2^k with forbidden subgroup \mathbb 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 \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≤ n/6, (and more generally, into any abelian group of order 2^k) is obtained from partitions of \mathbb F_2^m×\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.

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