A natural generalization of orthogonality of Latin squares

In this paper, a new concept, kk-plex orthogonality of Latin squares, is introduced. It generalizes the concept of orthogonality of Latin squares. Some examples of Latin squares with the new orthogonality are given. Bose, Shrikhande, and Parker’s Theorem is generalized to the case of kk-plex orthogonality for every positive integer kk while Mann’s Theorem is extended to the case of kk-plex orthogonality for every positive odd integer kk. Some other existence or nonexistence theorems are given. We also discuss constructions for kk-plex orthogonal Latin squares and generalize MacNeish’s Theorem.