Sets of mutually orthogonal Latin squares with “like subsquares”

AbstractThe concepts of “transversal identifying Latin squares” and Latin squares with “like subsquares” are introduced. The transversal identifying property permits the determination of orthogonality in computing time which is proportional to the cube of the order of the Latin square instead of the fourth power. Searching for sets of mutually orthogonal Latin squares with like subsquares also greatly reduces computing time. We have obtained orthogonal sets of sizes equal to the current lower bounds for n = 12, 15, 20, and 21 and have increased the lower bound for n = 24 from 4 to 5. In each of these orthogonal sets the Latin squares are all transversal identifying and all contain like subsquares