Amjadi, H., Soltankhah, N., Shajarisales, N., Tahvilian, M. (2016). On the spectrum of $r$-orthogonal Latin squares of different orders. Transactions on Combinatorics, 5(2), 41-51.

Hanieh Amjadi; Nasrin Soltankhah; Naji Shajarisales; Mehrdad Tahvilian. "On the spectrum of $r$-orthogonal Latin squares of different orders". Transactions on Combinatorics, 5, 2, 2016, 41-51.

Amjadi, H., Soltankhah, N., Shajarisales, N., Tahvilian, M. (2016). 'On the spectrum of $r$-orthogonal Latin squares of different orders', Transactions on Combinatorics, 5(2), pp. 41-51.

Amjadi, H., Soltankhah, N., Shajarisales, N., Tahvilian, M. On the spectrum of $r$-orthogonal Latin squares of different orders. Transactions on Combinatorics, 2016; 5(2): 41-51.

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

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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