University of Isfahan Transactions on Combinatorics 2251-8657 5 2 2016 06 01 On the spectrum of $r$-orthogonal Latin squares of different orders 41 51 11665 10.22108/toc.2016.11665 EN Hanieh Amjadi Alzahra University Nasrin Soltankhah Alzahra University Naji Shajarisales Max Planck Institute for Intelligent Systems Mehrdad Tahvilian Sharif University of Technology Journal Article 2015 04 27 ‎Two 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 $(3\leq n \leq 6)$‎, ‎the problem has been solved‎. https://toc.ui.ac.ir/article_11665_93148cb85b3fdaf4b4abfe8412331040.pdf