Quasi-Orthogonal Latin Squares and Related Designs

Abstract

We introduce a generalisation of the concept of a complete mapping of a group, which we call a quasi-complete mapping, and which leads us to a generalised form of orthogonality in Latin squares. In particular, the existence of a quasi-complete mapping of a group is shown to be sufficient for the existence of a pair of Latin squares such that if they are superimposed so as to form an array of unordered pairs, each unordered pair of distinct elements occurs exactly twice. We call such a pair of Latin squares quasi-orthogonal and prove that an abelian group possesses a quasi-complete mapping if and only if it is not of the form ${\mathbb{Z}}_{4m+2}\oplus G$, $|G|$ odd. In developing the theory of quasi-complete mappings, we show that the well-known concept of a quasi-complete Latin square arises quite naturally in this setting. We end the paper by giving a sufficient condition for the existence of a pair of quasi-orthogonal Latin squares which are also quasi-row-complete.