In this paper, we study the dual of generalized bent functions $f: V_{n}\rightarrow \mathbb{Z}_{p^k}$ where $V_{n}$ is an $n$-dimensional vector space over $\mathbb{F}_{p}$ and $p$ is an odd prime, $k$ is a positive integer. It is known that weakly regular generalized bent functions always appear in pairs since the dual of a weakly regular generalized bent function is also a weakly regular generalized bent function. The dual of non-weakly regular generalized bent functions can be generalized bent or not generalized bent. By generalizing the construction of \cite{Cesmelioglu5}, we obtain an explicit construction of generalized bent functions for which the dual can be generalized bent or not generalized bent. We show that the generalized indirect sum construction method given in \cite{Wang} can provide a secondary construction of generalized bent functions for which the dual can be generalized bent or not generalized bent. By using the knowledge on ideal decomposition in cyclotomic field, we prove that $f^{**}(x)=f(-x)$ if $f$ is a generalized bent function and its dual $f^{*}$ is also a generalized bent function. For any non-weakly regular generalized bent function $f$ which satisfies that $f(x)=f(-x)$ and its dual $f^{*}$ is generalized bent, we give a property and as a consequence, we prove that there is no self-dual generalized bent function $f: V_{n}\rightarrow \mathbb{Z}_{p^k}$ if $p\equiv 3 \ (mod \ 4)$ and $n$ is odd. For $p \equiv 1 \ (mod \ 4)$ or $p\equiv 3 \ (mod \ 4)$ and $n$ is even, we give a secondary construction of self-dual generalized bent functions. In the end, we characterize the relations between the generalized bentness of the dual of generalized bent functions and the bentness of the dual of bent functions, as well as the self-duality relations between generalized bent functions and bent functions by the decomposition of generalized bent functions.

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关于广义弯曲函数的对偶

在本文中，我们研究广义弯曲函数 $f 的对偶： V_{n}\rightarrow \mathbb{Z}_{p^k}$ 其中 $V_{n}$ 是 $n$ 维向量空间$\mathbb{F}_{p}$ 和 $p$ 是奇素数，$k$ 是正整数。众所周知，弱正则广义弯曲函数总是成对出现，因为弱正则广义弯曲函数的对偶也是弱正则广义弯曲函数。非弱正则广义弯曲函数的对偶可以是广义弯曲或非广义弯曲。通过推广 \cite{Cesmelioglu5} 的构造，我们获得了广义弯曲函数的显式构造，其中对偶可以是广义弯曲或非广义弯曲。我们表明 \cite{Wang} 中给出的广义间接和构造方法可以提供广义弯曲函数的二次构造，其中对偶可以是广义弯曲或非广义弯曲。利用分圆域中理想分解的知识，我们证明如果 $f$ 是广义弯函数及其对偶 $f^{*}$，则 $f^{**}(x)=f(-x)$也是广义弯曲函数。对于任何满足 $f(x)=f(-x)$ 及其对偶 $f^{*}$ 是广义弯曲的非弱正则广义弯曲函数 $f$，我们给出一个性质，因此，我们证明不存在自对偶广义弯曲函数 $f: V_{n}\rightarrow \mathbb{Z}_{p^k}$ if $p\equiv 3 \ (mod \ 4)$ and $n$很奇怪。对于 $p \equiv 1 \ (mod \ 4)$ 或 $p\equiv 3 \ (mod \ 4)$ 和 $n$ 是偶数，我们给出了自对偶广义弯曲函数的二次构造。最后，我们通过广义弯函数的分解刻画了广义弯函数对偶的广义弯度和弯函数对偶弯度之间的关系，以及广义弯函数和弯函数之间的自对偶关系。职能。