具有循环共生矩阵的部分几何设计
Partial geometric designs having circulant concurrence matrices
摘要：
我们调查部分几何设计，并研究其点的共存。设计的并发矩阵是对点对并发的编码，在一定程度上可以用于设计的分类。一个普通的2- $(v,k,λ)$设计对任意一对不同的点都有并发$λ$，并且它的并发矩阵是循环的.部分几何有两个并发$1$和$0$，横截设计TD $_λ(k, u)$有两个并发$λ$和$0$。还已知部分几何设计的共点矩阵最多可以有三个不同的特征值，且都是非负整数。本文证明了具有两个或三个不同并发的部分几何设计的存在性，并研究了哪些对称循环矩阵可以作为部分几何设计的并发矩阵。我们收集了已知的部分几何设计的来源，并研究其结构特征和构造方法。然后，我们给出了一个列表的可行的参数集的部分几何设计的顺序高达12，其中每个有一个循环的并发矩阵。我们还考虑了满足这些参数集的部分几何设计的组合性质和构造。
英文摘要：
We survey partial geometric designs and investigate their concurrences of points. The concurrence matrix of a design, which encodes the concurrences of pairs of points, can be used in the classification of designs in some extent. An ordinary 2-$(v,k,λ)$ design has concurrence $λ$ for any pair of distinct points, and its concurrence matrix is circulant. A partial geometry has two concurrences $1$ and $0$ and a transversal design TD$_λ(k, u)$ has two concurrences $λ$ and $0$. It is also known that the concurrence matrix of a partial geometric design can have at most three distinct eigenvalues, all of which are non-negative integers. In this paper, we show the existence of other partial geometric designs having two or three distinct concurrences, and investigate which symmetric circulant matrices are realized as the concurrence matrices of partial geometric designs. We collect known sources of partial geometric designs and study their structural characteristics and construction methods. We then give a list of feasible parameter sets for partial geometric designs of order up to 12 each of which has a circulant concurrence matrix. We also consider the combinatorial properties and constructions of partial geometric designs satisfying these parameter sets.
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