on the spectrum of r-orthogonal latin squares of different orders

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