Partial Spread and Vectorial Generalized Bent Functions

TL;DR

This paper extends the theory of generalized bent functions by describing partial spread classes, introducing vectorial gbent functions, and exploring their maximal dimensions and connections to difference sets.

Contribution

It generalizes the partial spread class for gbent functions, introduces vectorial gbent functions, and analyzes their maximal dimensions and relations to difference sets.

Findings

Complete description of gbent functions from _2^n to bZ_{2^t}

Introduction of vectorial gbent functions from _2^n to bZ_q^m

Determination of maximal m for q=2^t and relation to difference sets

Abstract

In this paper we generalize the partial spread class and completely describe it for generalized Boolean functions from $\F_2^n$ to $\mathbb{Z}_{2^t}$. Explicitly, we describe gbent functions from $\F_2^n$ 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 $\F_2^n$ to $\Z_q^m$, 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.