Nonexistence results of generalized bent functions from $\mathbb{Z}_3^n$ to $ \mathbb{Z}_m$

TL;DR

This paper explores the existence and nonexistence of generalized bent functions from ^n to m, establishing specific conditions under which such functions do or do not exist.

Contribution

It provides new nonexistence results for GBFs for certain dimensions and moduli, extending understanding of their structural limitations.

Findings

GBFs exist when 3 divides m

No GBFs for n=1,2 with odd m not divisible by 3

Nonexistence of GBFs for n=3 with certain moduli

Abstract

In this paper, we investigate generalized bent functions (GBFs) from $\mathbb{Z}_3^n$ to $\mathbb{Z}_m$. We show that GBFs exist whenever $3$ divides $m$, while several nonexistence results are obtained when $3\nmid m$. In particular, we prove that no GBFs exist for $n=1,2$ when $m$ is odd and not divisible by $3$. For the case $n=3$, we establish the nonexistence of GBFs $f:\mathbb{Z}_3^3 \rightarrow \mathbb{Z}_{5\cdot11^r}$ for all nonnegative integers $r$. Finally, we show that no GBF exists from $\mathbb{Z}_3$ to $\mathbb{Z}_{2m'}$ and $\mathbb{Z}_3^2$ to $\mathbb{Z}_{2m'}$, where $m'$ is odd and not divisible by $3$.