There are two construction methods of designs from $(n,m)$ -bent functions, known as translation and addition designs. In this article we analyze, which equivalence relation for Boolean bent functions, i.e. $(n,1)$ -bent functions, and vectorial bent functions, i.e. $(n,m)$ -bent functions with $2\le m\le n/2$ , is coarser: extended-affine equivalence or isomorphism of associated translation and addition designs. First, we observe that similar to the Boolean bent functions, extended-affine equivalence of vectorial $(n,m)$ -bent functions and isomorphism of addition designs are the same concepts for all even $n$ and $m\le n/2$ . Further, we show that extended-affine inequivalent Boolean bent functions in $n$ variables, whose translation designs are isomorphic, exist for all $n\ge 6$ . This implies, that isomorphism of translation designs for Boolean bent functions is a coarser equivalence relation than extended-affine equivalence. However, we do not observe the same phenomenon for vectorial bent functions in a small number of variables. We classify and enumerate all vectorial bent functions in six variables and show, that in contrast to the Boolean case, one cannot exhibit isomorphic translation designs from extended-affine inequivalent vectorial $(6,m)$ -bent functions with $m\in \{ 2,3 \}$ .

中文翻译：

---

布尔函数和向量弯曲函数的设计理论方面

设计的构造方法有两种 $(n,m)$ -bent 函数，称为平移和加法设计。在本文中，我们分析了布尔弯曲函数的等价关系，即 $(n,1)$ -bent 函数和矢量弯曲函数，即 $(n,m)$ -弯曲的功能与 $2\le m\le n/2$ , 更粗略：相关翻译和加法设计的扩展仿射等价或同构。首先，我们观察到类似于布尔弯曲函数，向量的扩展仿射等价 $(n,m)$ -加法设计的弯曲函数和同构对于所有偶数来说都是相同的概念 $n$ 和 $m\le n/2$ . 此外，我们证明了扩展仿射不等价布尔弯曲函数 $n$ 变量，其翻译设计是同构的，存在于所有 $n\ge 6$ . 这意味着，布尔弯曲函数的翻译设计的同构是比扩展仿射等价更粗略的等价关系。然而，对于少量变量中的向量弯曲函数，我们没有观察到相同的现象。我们对六个变量中的所有矢量弯曲函数进行分类和枚举，并表明，与布尔情况相比，不能展示来自扩展仿射不等价矢量的同构翻译设计 $(6,m)$ -弯曲的功能与 $m\in \{ 2,3 \}$ .