Nonexistence of certain classes of generalized bent functions: Revisiting the element partition method

TL;DR

This paper proves new nonexistence results for certain classes of generalized bent functions using an extended element partition method, focusing on specific prime factorizations.

Contribution

It extends the element partition method to establish nonexistence of generalized bent functions for new parameter classes.

Findings

Nonexistence of generalized bent functions for q=2 p_1^{e_1} p_2^{e_2} with specific primes.

Proved nonexistence of functions of type [1, 2·3^a·7^b] for positive integers a, b.

Extended the element partition method to broader classes of generalized bent functions.

Abstract

We obtain new nonexistence results of two classes of generalized bent functions from $\mathbb{Z}_{q}^{n}$ to $\mathbb{Z}_{q}$ (called type $[n,q]$). The first class of results is based on applying the element partition method to the results of Feng and Feng and Liu, where $q=2 p_1^{e_1} p_{2}^{e_2}$, $p_1$ and $p_2$ are two primes. For the second class, we extend the idea of the element partition method and prove the nonexistence of generalized bent functions of type $[1,2 \cdot 3^{a} \cdot 7^{b}]$, where $a,b \in \mathbb{Z}_{>0}$.