@article {
author = {Amjadi, Hanieh and Soltankhah, Nasrin and Shajarisales, Naji and Tahvilian, Mehrdad},
title = {On the spectrum of $r$-orthogonal Latin squares of different orders},
journal = {Transactions on Combinatorics},
volume = {5},
number = {2},
pages = {41-51},
year = {2016},
publisher = {University of Isfahan},
issn = {2251-8657},
eissn = {2251-8665},
doi = {10.22108/toc.2016.11665},
abstract = {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.},
keywords = {Latin square,Orthogonal Latin square,$r$-Orthogonal Latin square,$r$-Orthogonality spectrum,Transversal},
url = {https://toc.ui.ac.ir/article_11665.html},
eprint = {https://toc.ui.ac.ir/article_11665_93148cb85b3fdaf4b4abfe8412331040.pdf}
}