AI 中文总结
研究推广(k,n)-网等的组合结构，通过网状结构定义协作系统与半正交阵列，引入副同构和同构概念，给出小网状结构构造及大网状结构的三种递归构造。
AI 中文摘要
我们研究了推广(k,n)-网、正交阵列和相互正交拉丁方的组合结构。通过网状结构，我们指的是一个点集以及两类线族，使得不同类型的两条线恰好在一个点相交，且每个线族划分点集。与任何线关联的点的数量仅取决于线的类型，并且每个点与给定类型的相同数量的线关联。每种类型的一个线族的每个选择都会导致点排列成矩形网格。在阵列的相应位置记录包含给定点的线会得到相互正交拉丁方集的一种推广，称为协作系统。协作系统由列拉丁矩阵集合和行拉丁矩阵集合组成，使得每个列拉丁矩阵与每个行拉丁矩阵正交。将包含给定点的线记录为元组会得到某些正交阵列的一种推广，称为半正交阵列。引入了协作系统的副同构和同构概念，分别对应于置换线族和每个族内的线。给出了小网状结构的构造以及大网状结构的三种递归构造。
英文摘要
We examine combinatorial structures that generalize $(k,n)$-nets, orthogonal arrays, and mutually orthogonal Latin squares. By a reticulation we mean a point set and two collections (types) of families of lines such that two lines of different types meet in exactly one point and each family of lines partitions the point set. The number of points incident with any line depends only upon the type of the line, and every point is incident with the same number of lines of a given type. Each choice of a single line family of each type leads to an arrangement of the points into a rectangular grid. Recording the line containing a given point in the corresponding position of an array gives a generalization of sets of mutually orthogonal Latin squares, dubbed a cooperative system. A cooperative system consists of a collection of column-Latin matrices and a collection of row-Latin matrices such that each column-Latin matrix is orthogonal to each row-Latin matrix. Recording lines which contain a given point as a tuple gives a generalization of certain orthogonal arrays, dubbed semi-orthogonal arrays. Notions of parastrophy and isotopy for cooperative systems, corresponding to permuting families of lines and lines within each family, are introduced. Constructions of small reticulations and three recursive constructions of larger reticulations are given.
CommentsV2 update reference with arXiv link for for companion paper; grammatical corrections; reformat front matter. 33 pages
