关于模平衡划分设计
On modular balanced partition designs
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中文总结 AI 辅助
研究有限整数集\(X\)的模平衡划分设计，通过证明存在条件、确定与子群幻方关系、利用阿贝尔群仿射自同构证明其存在性，并提供构造横截设计的方法，为相关设计理论提供了新内容。
中文摘要 AI 辅助
设\(X\)是一个基数为\(\nu=\kappa\lambda\)的有限整数集。一个模平衡划分设计是一个三元组\((X,\mathcal{A},\mathcal{B})\)，满足以下条件：\(\mathcal{A}\)是\(X\)的一个划分，分成\(\kappa\)个大小为\(\lambda\)的块，且\(X\)的每个元素恰好出现在一个块中；\(\mathcal{B}\)是\(X\)的一个划分，分成\(\lambda\)个大小为\(\kappa\)的块，且\(X\)的每个元素恰好出现在一个块中；\(A_i\cap B_j\)恰好有一个元素。我们证明了模平衡划分设计存在的必要条件，研究并确定了它与子群幻方的关系，利用阿贝尔群的仿射自同构证明了非同构模平衡划分设计的存在性，最后提供了一种通过模平衡划分设计构造横截设计的方法。
英文摘要
Let $X$ be a finite set of integers with cardinality $ν= κλ$. A \emph{modular balanced partition design} is a triplet $(X, \mathcal{A}, \mathcal{B})$ satisfying the following conditions: \begin{itemize} \item $\mathcal{A}$ is a partition of $X$ into $κ$ blocks of size $λ$, such that every element of $X$ appears in exactly one block. If $\mathcal{A} = \{A_1, A_2, \ldots, A_κ\}$, then $\sum_{a\in A_i} a \equiv i λ\pmodν$, for each $i=1,2,\ldots,κ$ \item $\mathcal{B}$ is a partition of $X$ into $λ$ blocks of size $κ$, such that every element of $X$ appears in exactly one block. If $\mathcal{B} = \{B_1, B_2, \ldots, B_λ\}$, then $\sum_{b\in B_j} b \equiv j κ\pmodν$, for each $j=1,2,\ldots,λ$ \item $A_i \cap B_j$ has exactly one element, for any $A_i \in \mathcal{A}$, and $B_j \in \mathcal{B}$. \itemize} We prove the necessary conditions for the existence of a modular balanced partition design. Moreover, we investigate and identify a relationship between a modular balanced partition design and a subgroup magic rectangle. Then by using affine automorphisms of an Abelian group, we prove the existence of non-isomorphic modular balanced partition designs. Finally, we provide a method to construct a transversal design via a modular balanced partition design.
