Orthogonal Steiner triple systems of order \(6m + 3\)

In this paper, we prove that for any \(n > 27363\), \(n \equiv 3\) modulo {6}, there exist a pair of orthogonal Steiner triple systems of order \(n\). Further, a pair of orthogonal Steiner triple systems of order \(n\) exist for all \(n \equiv 3\) modulo {6}, {3} \(< n \leq 27363\), with at most \(918\) possible exceptions. The proof of this result depends mainly on the construction of pairwise balanced designs having block sizes that are prime powers congruent to \(1\) modulo {6}, or \(15\) or \(27\). Some new examples are also constructed recursively by using conjugate orthogonal quasigroups.