More characterizations of generalized bent function in odd characteristic, their dual and the gray image

TL;DR

This paper explores properties of generalized bent functions over odd prime fields, introduces a subclass linked to difference sets, and analyzes their duals and Gray images, extending to functions with different domain structures.

Contribution

It provides new characterizations of gbent functions, introduces the concept of ent functions related to difference sets, and extends the theory to functions with varied domain parameters.

Findings

Characterizations of gbent functions in terms of component functions

Introduction of ent functions related to relative difference sets

Analysis of duals and Gray images of gbent functions

Abstract

In this paper, we further investigate properties of generalized bent Boolean functions from $\Z_{p}^n$ to $\Z_{p^k}$, where $p$ is an odd prime and $k$ is a positive integer. For various kinds of representations, sufficient and necessary conditions for bent-ness of such functions are given in terms of their various kinds of component functions. Furthermore, a subclass of gbent functions corresponding to relative difference sets, which we call $\Z_{p^k}$-bent functions, are studied. It turns out that $\Z_{p^k}$-bent functions correspond to a class of vectorial bent functions, and the property of being $\Z_{p^k}$-bent is much stronger then the standard bent-ness. The dual and the generalized Gray image of gbent function are also discussed. In addition, as a further generalization, we also define and give characterizations of gbent functions from $\Z_{p^l}^n$ to $\Z_{p^k}$ for a positive integer $l$ with $l<k$.