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 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 ≥ 2n 3 . Here for k < 2n 3 , some constructions are shown to realize many values of r and for small cases (3 ≤ n ≤ 6), the problem has been solved.