On opposite orthogonal Steiner triple systems of non-prime-power order

Abstract

Consider Steiner triple systems (STS) developed from a Steiner difference family in an additive abelian group. The opposite of such an STS is formed by taking the negative of each triple (elementwise). Two STS( n) are said to be orthogonal if their sets of triples are disjoint, and two disjoint pairs defining intersecting triples in one system fail to do so in the other. It is well-known that orthogonal STS naturally give rise to a Room square; when it is skew, the pair of STS is called skew-orthogonal. A known field construction shows that all prime power orders n≡1 ( mod 6) admit STS orthogonal (in fact, skew-orthogonal) to their opposites. It is noted in this paper that an infinite family of non-prime-power orders do not admit STS skew-orthogonal to their opposites. However, computational methods easily find STS orthogonal to their opposites in cyclic groups of all 1 ( mod 6) orders n<1000, n ≠ 25 . In particular, this settles the results of a search for order 55 mentioned in an earlier paper of Schreiber.