广义Bent函数奇特征、对偶特征及灰度图像的进一步刻画
More characterizations of generalized bent function in odd characteristic, their dual and the gray image
摘要：
本文进一步研究了从$\Z_{p}^n$到$\Z_{p^k}$的广义Bent布尔函数的性质，其中$p$是奇素数，$k$是正整数.对于各种表示，根据它们的各种分支函数，给出了这类函数为非线性的充分条件和必要条件。进一步研究了与相对差集对应的gbent函数的一个子类，我们称之为$\Z_{p^k}$ -bent函数.结果表明，$\Z_{p^k}$ -bent函数对应于一类向量bent函数，且其$\Z_{p^k}$ -bent性质比标准bent性质强得多.讨论了gbent函数的对偶和广义Gray象。此外，作为进一步的推广，我们还定义并给出了从$\Z_{p^l}^n$到$\Z_{p^k}$的gbent函数的特征，其中$l$为正整数，且具有$l<k$ .
英文摘要：
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 $l<k$.
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