New and Old Values for Maximal MOLS\((n)\)

Let \(k\) Max MOLS\((n)\) denote a maximal set of \(k\) mutually orthogonal Latin squares of order \(n\), and let the parameter triple \((G,n,k)\) denote the existence of a \(k\) Max MOLS\((n)\) constructed from orthogonal orthomorphisms of a group \(G\) of order \(x\). We identify all such parameter triples for all \(G\) of order \(\leq 15\), and report the existence of \(3\) Max MOLS\((n)\) for \(n = 15, 16\) and \(4\) Max MOLS\((n)\) for \(n = 12, 16, 24, 28\). Our work shows that for \(n \leq 15\), all known parameter pairs \((n, k)\) for which there exists a \(k\) Max MOLS\((n)\) can be attained by constructing maximal sets of MOLS from orthomorphisms of groups, except for \(1\) Max MOLS\((n)\), \(n = 5, 7, 9, 13\) and \(2\) Max MOLS\((10)\).