A few more r-orthogonal latin squares

AbstractTwo 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⩽n2, and an easy argument establishes that r∉{n+1,n2−1}. In a recent paper by Colbourn and Zhu, this question has been answered leaving only a few possible exceptions for r=n2−3 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 (142−3)-orthogonal Latin squares