Partial spread and vectorial generalized bent functions

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.