A few more \(r\)-orthogonal Latin squares (Q5943896)

Two Latin squares of order \(n\) are said to be \(r\)-orthogonal if upon superposition, exactly \(r\) distinct ordered pairs occur. Thus for Latin squares of order \(n\), it is clear that \(n \leq r \leq n^2\). A well-studied question is for a given value of \(n\), what values of \(r\) can occur? This problem had earlier been resolved except for a few values of \(n\) and in this note it is shown that there exist \(r\)-orthogonal Latin squares of order \(n\) for \(n\leq r \leq n^2\), \(r\not=n+1\), \(n^2-1\) apart from the genuine exceptions: \(n=2\), \(r=4\); \(n=3\), \(r=5,6,7\); \(n=4\), \(r=7,10,11, 13,14\); \(n=5\), \(r=8,9,20,22,23\); \(n=6\), \(r=33,36\). The only unresolved case occurs when \(n=14\) and \(r=n^2-3 =193\).