简单正交多阵SOMA(4,n)的存在性

Existence of Simple Orthogonal Multi-array SOMA(4, n)

设整数k>0,n≥2,简单正交多阵SOMA(k,n)是一个n×n的方阵A,它的每一个单元格包含kn-元集S的一个k-元子集且满足:1)S的每个元素在A的每一行和每一列恰好出现一次;2)S的每个2-元子集至多出现在A的一个单元格中.总结了SOMA(k,n)存在的最新结果,并且阐述了它和相互正交拉丁方之间的关系,同时通过直接构造的方法证明了n≥5,SOMA(4,n)的存在性,并且提出了进一步研究的问题.

Let k 0 and n ≥ 2 be integers. A simple orthogonal multi-array SOMA（k, n） is an n × n array A each of whose entries is a k-subset of a kn-set S of symbols, such that every symbol of S occurs exactly once in each row and exactly once in each column of A, and every 2-subset of S is contained in at most one entry of A. In this note, it is summarized that the latest results of the existence of a SOMA（k, n） and explain the relationship between it and mutually orthogonal Latin squares, and by direct construction show that there exists a SOMA（4, n） for n ≥ 5.And further research questions are also put forward.