The Orthogonality Spectrum for Latin Squares of Different Orders

Two orthogonal latin squares of order n have the property that when they are superimposed, each of the n (2) ordered pairs of symbols occurs exactly once. In a series of papers, Colbourn, Zhu, and Zhang completely determine the integers r for which there exist a pair of latin squares of order n having exactly r different ordered pairs between them. Here, the same problem is considered for latin squares of different orders n and m. A nontrivial lower bound on r is obtained, and some embedding-based constructions are shown to realize many values of r.