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More characterizations of generalized bent function in odd characteristic, their dual and the gray image

In this paper, we further investigate properties of generalized bent Boolean functions from $\Z_{p}^n$ to $\Z_{p^k}$, where $p$ is an odd prime and $k$ is a positive integer. For various kinds of representations, sufficient and necessary conditions for bent-ness of such functions are given in terms of their various kinds of component functions. Furthermore, a subclass of gbent functions corresponding to relative difference sets, which we call $\Z_{p^k}$-bent functions, are studied. It turns out that $\Z_{p^k}$-bent functions correspond to a class of vectorial bent functions, and the property of being $\Z_{p^k}$-bent is much stronger then the standard bent-ness. The dual and the generalized Gray image of gbent function are also discussed. In addition, as a further generalization, we also define and give characterizations of gbent functions from $\Z_{p^l}^n$ to $\Z_{p^k}$ for a positive integer $l$ with $l<k$.