Sets of three pairwise orthogonal Steiner triple systems

Two Steiner triple systems (STS) are orthogonal if their sets of triples are disjoint, and two disjoint pairs of points defining intersecting triples in one system fail to do so in the other. In 1994, it was shown (Canad. J. Math. 46(2) (1994) 239–252) that there exist a pair of orthogonal Steiner triple systems of order v for all v≡1,3 (mod 6), with v⩾7, v≠9. In this paper we show that there exist three pairwise orthogonal Steiner triple systems of order v for all v≡1 (mod 6), with v⩾19 and for all v≡3 (mod 6), with v⩾27 with only 24 possible exceptions.