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

Abstract

We obtain new nonexistence results for two classes of generalized bent functions from $\mathbb{Z}_{q}^{n}$ to $\mathbb{Z}_{q}$, called type $[n,q]$ generalized bent functions. The first class concerns the case $q=2 p_1^{e_1} p_2^{e_2}$, where $p_1$ and $p_2$ are distinct odd primes. By applying the element partition method introduced by Lv and Li to earlier results of Feng and Feng-Liu, we obtain sharper nonexistence results for several families of parameters satisfying explicit congruence and order conditions. These results extend known nonexistence theorems in cases where the prime divisors of the odd part of $q$ are self-conjugate. The second class concerns the case $q=2 \cdot 3^a \cdot 7^b$. By extending the idea of the element partition method and combining it with explicit computations in suitable cyclotomic fields and their subfields, we prove that generalized bent functions of type $[1,2\cdot 3^a\cdot 7^b]$ do not exist for all positive integers $a$ and $b$.