On some properties of self-dual generalized bent functions

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. A gbent function for which it is possible to define a dual gbent function is called regular. A regular gbent function is said to be self-dual if it coincides with its dual. We obtain the necessary and sufficient conditions for the self-duality of gbent functions from Eliseev — Maiorana — McFarland class. We find the complete Lee distance spectrum between all self-dual functions in this class and obtain that the minimal Lee distance between them is equal to $q\cdot2^{n-3}$. For Boolean case, there are no affine bent functions and self-dual bent functions, while it is known that for generalized case affine bent functions exist, in particular, when $q$ is divisible by $4$. We prove the non-existence of affine self-dual gbent functions for any natural even $q$. A new class of isometries preserving self-duality of a gbent function is presented. Based on this, a refined classification of self-dual gbent functions of the form $\mathbb{F}_2^4\rightarrow\mathbb{Z}_4$ is given.