类展布划分的Bent和$\mathbb Z_{2^k}$ -Bent函数
Bent and $\mathbb Z_{2^k}$-bent functions from spread-like partitions
摘要：
从偶数维$n=2m$的$\mathbb F_2$上的向量空间$V_n$到循环群$\mathbb Z_{2^k}$的Bent函数，或者等价地，具有禁止子群$\mathbb Z_{2^k}$的$V_n\times\mathbb Z_{2^k}$中的相对差集，可以从对于任何$k\le n/2$的$V_n$的扩展中获得。本文研究了从$V_n$到$\mathbb Z_{2^k}$的Bent函数的存在性和构造性，这些函数不是由展开构造得到的.从$\mathbb F_{2^m}\times\mathbb F_{2^m}$的划分得到从$V_n$到$\mathbb Z_{2^k}$、$k\le n/6$（更一般地，到任何$2^k$阶的阿贝尔群）的Bent函数的构造，这可以被看作是Desklesian spread的推广。对于扩散，这些划分的一定固定数量的集合的并集总是布尔bent函数的支撑。
英文摘要：
Bent functions from a vector space $V_n$ over $\mathbb F_2$ of even dimension $n=2m$ into the cyclic group $\mathbb Z_{2^k}$, or equivalently, relative difference sets in $V_n\times\mathbb Z_{2^k}$ with forbidden subgroup $\mathbb Z_{2^k}$, can be obtained from spreads of $V_n$ for any $k\le n/2$. In this article, existence and construction of bent functions from $V_n$ to $\mathbb Z_{2^k}$, which do not come from the spread construction is investigated. A construction of bent functions from $V_n$ into $\mathbb Z_{2^k}$, $k\le n/6$, (and more generally, into any abelian group of order $2^k$) is obtained from partitions of $\mathbb F_{2^m}\times\mathbb F_{2^m}$, which can be seen as a generalization of the Desarguesian spread. As for the spreads, the union of a certain fixed number of sets of these partitions is always the support of a Boolean bent function.
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