University of IsfahanTransactions on Combinatorics2251-86575220160601On the spectrum of $r$-orthogonal Latin squares of different orders41511166510.22108/toc.2016.11665ENHanieh AmjadiAlzahra UniversityNasrin SoltankhahAlzahra UniversityNaji ShajarisalesMax Planck Institute for Intelligent SystemsMehrdad TahvilianSharif University of TechnologyJournal Article20150427Two Latin squares of order $n$ are orthogonal if in their superposition, each of the $n^{2}$ ordered pairs of symbols occurs exactly once. Colbourn, Zhang and Zhu, in a series of papers, determined the integers $r$ for which there exist a pair of Latin squares of order $n$ having exactly $r$ different ordered pairs in their superposition. Dukes and Howell defined the same problem for Latin squares of different orders $n$ and $n+k$. They obtained a non-trivial lower bound for $r$ and solved the problem for $k geq frac{2n}{3} $. Here for $k < frac{2n}{3}$, some constructions are shown to realize many values of $r$ and for small cases $(3leq n leq 6)$, the problem has been solved.http://toc.ui.ac.ir/article_11665_93148cb85b3fdaf4b4abfe8412331040.pdf