Computing Random r-Orthogonal Latin Squares

Two Latin squares of order n are r-orthogonal if, when superimposed, there are exactly r distinct ordered pairs. The spectrum of all values of r for Latin squares of order n is known. A Latin square A of order n is r-self-orthogonal if A and its transpose are r-orthogonal. The spectrum of all values of r is known for all orders \(n\ne 14\) . We develop randomized algorithms for computing pairs of r-orthogonal Latin squares of order n and algorithms for computing r-self-orthogonal Latin squares of order n.