A few more r-orthogonal Latin squares

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 less than or equal tor less than or equal ton(2), and an easy argument establishes that r is not an element of {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 is an element of {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. (C) 2001 Elsevier Science B.V. All rights reserved.