Partial spread and vectorial generalized bent functions

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.