TY - JOUR
ID - 11665
TI - On the spectrum of $r$-orthogonal Latin squares of different orders
JO - Transactions on Combinatorics
JA - TOC
LA - en
SN - 2251-8657
AU - Amjadi, Hanieh
AU - Soltankhah, Nasrin
AU - Shajarisales, Naji
AU - Tahvilian, Mehrdad
AD - Alzahra University
AD - Max Planck Institute for Intelligent Systems
AD - Sharif University of Technology
Y1 - 2016
PY - 2016
VL - 5
IS - 2
SP - 41
EP - 51
KW - Latin square
KW - Orthogonal Latin square
KW - $r$-Orthogonal Latin square
KW - $r$-Orthogonality spectrum
KW - Transversal
DO - 10.22108/toc.2016.11665
N2 - 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 $(3leq n leq 6)$, the problem has been solved.
UR - https://toc.ui.ac.ir/article_11665.html
L1 - https://toc.ui.ac.ir/article_11665_93148cb85b3fdaf4b4abfe8412331040.pdf
ER -