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

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 $f:{\mathbb{Z}}_3^3\to {\mathbb{Z}}_{5\cdot 11^r}$ for all nonnegative integers $r$. Finally, we show that no GBF exists from ${\mathbb{Z}}_3$ to ${\mathbb{Z}}_{2m^{\prime }}$ and ${\mathbb{Z}}_3^2$ to ${\mathbb{Z}}_{2m^{\prime }}$, where $m^{\prime }$ is odd and not divisible by $3$.