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

**ZHANG Yanyan;LI Yun;WANG Su;School of Sciences,Nantong University;Basic Teaching Institute,Nantong Institute of Technology;Nantong University Xinglin College;**

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.

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