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^{۲} 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{۲n}{۳} ‎. ‎Here for k < \frac{۲n}{۳}‎, ‎some constructions are shown to realize many values of r and for small cases (۳\leq n \leq ۶)‎, ‎the problem has been solved‎.

Keywords:

‎Latin square‎ , ‎Orthogonal Latin square‎ , ‎‎‎r‎‎-Orthogonal Latin square‎ , ‎‎‎r‎‎-Orthogonality spectrum‎ , ‎Transversal