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 ${\text{F}}_2^n$ to ${\mathbb{Z}}_{2^t}$. Explicitly, we describe gbent functions from ${\text{F}}_2^n$ to ${\mathbb{Z}}_{2^t}$, which can be seen as a gbent version of Dillon’s $P{S}_{ap}$ class. For the first time, we also introduce the concept of a vectorial gbent function from ${\text{F}}_2^n$ 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.