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Bent and $\mathbb Z_{2^k}$-bent functions from spread-like partitions (2009.11019v1)

Published 23 Sep 2020 in math.NT, cs.IT, and math.IT

Abstract: Bent functions from a vector space $V_n$ over $\mathbb F_2$ of even dimension $n=2m$ into the cyclic group $\mathbb Z_{2k}$, or equivalently, relative difference sets in $V_n\times\mathbb Z_{2k}$ with forbidden subgroup $\mathbb Z_{2k}$, 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 $\mathbb Z_{2k}$, which do not come from the spread construction is investigated. A construction of bent functions from $V_n$ into $\mathbb Z_{2k}$, $k\le n/6$, (and more generally, into any abelian group of order $2k$) is obtained from partitions of $\mathbb F_{2m}\times\mathbb F_{2m}$, 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.

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