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.