On the Structure and Classification of SOMAs: Generalizations of Mutually Orthogonal Latin Squares

Abstract

Let $k\ge0$ and $n\ge2$ be integers. A SOMA, or more specifically a SOMA$(k,n)$, is an $n\times n$ array $A$, whose entries are $k$-subsets of a $kn$-set $\Omega$, such that each element of $\Omega$ occurs exactly once in each row and exactly once in each column of $A$, and no 2-subset of $\Omega$ is contained in more than one entry of $A$. A SOMA$(k,n)$ can be constructed by superposing $k$ mutually orthogonal Latin squares of order $n$ with pairwise disjoint symbol-sets, and so a SOMA$(k,n)$ can be seen as a generalization of $k$ mutually orthogonal Latin squares of order $n$. In this paper we first study the structure of SOMAs, concentrating on how SOMAs can decompose. We then report on the use of computational group theory and graph theory in the discovery and classification of SOMAs. In particular, we discover and classify SOMA$(3,10)$s with certain properties, and discover two SOMA$(4,14)$s (SOMAs with these parameters were previously unknown to exist). Some of the newly discovered SOMA$(3,10)$s come from superposing a Latin square of order 10 on a SOMA$(2,10)$.