具有$λ_1=3$和大第二指标的分组可分设计
Group Divisible Designs with $λ_1=3$ and Large Second Index
摘要：
可分组设计$\mbox{GDD}(m,n;λ_1,λ_2)$是有序对$(V, \cal{B})$，其中$V$是符号的$(m+n)$ -集合，而$\cal{B}$是$3$ -子集的集合将$(m+n)$ -集合分成大小为$m$和大小为$n$的2个组：来自同一组的每对符号恰好出现在$\cal{B}$中的$λ_1$块中，而来自不同组的每对符号恰好出现在$\cal{B}$中的$λ_2$块中。$λ_1$和$λ_2$分别被称为第一索引和第二索引。在这里，我们专注于$\mbox{GDD}$的存在性问题时，$λ_1=3$和$λ_2>3$。我们得到的必要条件，并证明这些条件是充分的大多数情况下。
英文摘要：
A group divisible design $\mbox{GDD}(m,n;λ_1,λ_2)$, is an ordered pair $(V, \cal{B})$ where $V$ is an $(m+n)$-set of symbols while $\cal{B}$ is a collection of $3$-subsets (called blocks) of $V$ satisfying the following properties: the $(m+n)$-set is divided into 2 groups of size $m$ and of size $n$: each pair of symbols from the same group occurs in exactly $λ_1$ blocks in $\cal{B}$, and each pair of symbols from different groups occurs in exactly $λ_2$ blocks in $\cal{B}$. $λ_1$ and $λ_2$ are referred to as first index and second index, respectively. Here, we focus on an existence problem of $\mbox{GDD}$s when $λ_1=3$ and $λ_2>3$. We obtain the necessary conditions and prove that these conditions are sufficient for most of the cases.
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