Full characterization of generalized bent functions as (semi)-bent spaces, their dual, and the Gray image

TL;DR

This paper provides a comprehensive algebraic characterization of generalized bent functions for all powers of two, including their duals and Gray images, and explores subclasses related to difference sets and vectorial bent functions.

Contribution

It offers a complete description of gbent functions as affine spaces of bent or semi-bent functions for all q=2^k, extending previous results to all n and q.

Findings

Characterization of gbent functions as affine spaces with specific properties

Explicit description of dual and Gray image of gbent functions

Identification of a subclass related to difference sets and vectorial bent functions

Abstract

In difference to many recent articles that deal with generalized bent (gbent) functions $f:\mathbb{Z}_2^n \rightarrow \mathbb{Z}_q$ for certain small valued $q\in \{4,8,16 \}$, we give a complete description of these functions for both $n$ even and odd and for any $q=2^k$ in terms of both the necessary and sufficient conditions their component functions need to satisfy. This enables us to completely characterize gbent functions as algebraic objects, namely as affine spaces of bent or semi-bent functions with interesting additional properties, which we in detail describe. We also specify the dual and the Gray image of gbent functions for $q=2^k$. We discuss the subclass of gbent functions which corresponds to relative difference sets, which we call $\mathbb{Z}_q$-bent functions, and point out that they correspond to a class of vectorial bent functions. The property of being $\mathbb{Z}_q$-bent is much stronger than the standard concept of a gbent function. We analyse two examples of this class of functions.