Generalizations of nets and Latin squares

Abstract

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.