A few more r-orthogonal latin squares

Abstract Two Latin squares are r-orthogonal if their superposition produces r distinct pairs. It was Belyavskaya who first systematically treated the following question: For which integers n and r does a pair of r-orthogonal Latin squares of order n exist? Evidently, n ⩽ r ⩽ n 2 , and an easy argument establishes that r ∉{ n +1, n 2 −1}. In a recent paper by Colbourn and Zhu, this question has been answered leaving only a few possible exceptions for r = n 2 −3 and n ∈{6,7,8,10,11,13,14,16,17,18,19,20,22,23,25,26}. In this paper, these possible exceptions are removed by direct and recursive constructions except two orders n =6,14. For n =6, a computer search shows that r =33 is a genuine exception. For n =14, it is still undecided if there exists a pair of (14 2 −3)-orthogonal Latin squares.