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

‎ Two Latin squares of order nn are orthogonal if in their superposition‎, ‎each of the n2 ordered pairs of symbols occurs exactly once‎. ‎Colbourn‎, ‎Zhang and Zhu‎, ‎in a series of papers‎, ‎determined the integers rr for which there exist a pair of Latin squares of order nn having exactly r different ordered pairs in their superposition‎. ‎Dukes and Howell defined the same problem for Latin squares of different orders nn and n+kn+k‎. ‎They obtained a non-trivial lower bound for rr and solved the problem for k≥2n3‎. ‎Here for k<2n3‎, ‎some constructions are shown to realize many values of rr and for small cases (3≤n≤6)‎, ‎the problem has been solved‎.