On constructions and properties of self-dual generalized bent functions

TL;DR

This paper investigates the construction and properties of self-dual generalized bent functions, providing new classifications, conditions, and symmetry analyses for these functions, especially for even q.

Contribution

It introduces new constructions and characterizations of self-dual gbent functions, including necessary and sufficient conditions and symmetry properties.

Findings

Number of self-dual and anti-self-dual gbent functions coincide

Self-duality conditions for Maiorana--McFarland gbent functions are established

Self-dual gbent functions cannot be affine

Abstract

Bent functions of the form $\mathbb{F}_2^n\rightarrow\mathbb{Z}_q$, where $q\geqslant2$ is a positive integer, are known as generalized bent (gbent) functions. Gbent functions for which it is possible to define a dual gbent function are called regular. A regular gbent function is said to be self-dual if it coincides with its dual. In this paper we explore self-dual gbent functions for even $q$. We consider several primary and secondary constructions of such functions. It is proved that the numbers of self-dual and anti-self dual gbent functions coincide. We give necessary and sufficient conditions for the self-duality of Maiorana--McFarland gbent functions and find Hamming and Lee distances spectrums between them. We find all self-dual gbent functions symmetric with respect to two variables and prove that self-dual gbent function can not be affine. The properties of sign functions of self-dual gbent functions are considered. Symmetries that preserve self-duality are also discussed.