Partial spread and vectorial generalized bent functions
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Abstract
In this paper we generalize the partial spread class and completely describe it for generalized Boolean functions from \({\mathbb {F}}_2^n\) to \({\mathbb {Z}}_{2^t}\). Explicitly, we describe gbent functions from \({\mathbb {F}}_2^n\) to \({\mathbb {Z}}_{2^t}\), which can be seen as a gbent version of Dillon’s \(PS_{ap}\) class. For the first time, we also introduce the concept of a vectorial gbent function from \({\mathbb {F}}_2^n\) to \({\mathbb {Z}}_q^m\), and determine the maximal value which m can attain for the case \(q=2^t\). Finally we point to a relation between vectorial gbent functions and relative difference sets.
Keywords
Generalized Boolean function Generalized bent function Partial spread Vectorial function Relative difference setMathematics Subject Classification
94C10 06E30 05B10Notes
Acknowledgments
Work by P. S. started during a very enjoyable visit at RICAM (Johann Radon Institute for Computational and Applied Mathematics), Austrian Academy of Sciences, in Linz, Austria. Both the second and third named author thank the institution for the excellent working conditions. The second author is supported by the Austrian Science Fund (FWF) Project No. M 1767-N26.
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