# On the spectrum of $r$-orthogonal Latin squares of different orders

Document Type: Research Paper

Authors

1 Alzahra University

2 Max Planck Institute for Intelligent Systems

3 Sharif University of Technology

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.

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### References

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