Sets of three pairwise orthogonal Steiner triple systems (Q1865412)

This paper does a nice job in extending the work of \textit{C. J. Colbourn, P. B. Gibbons, R. Mathon, R. C. Mullin} and \textit{A. Rosa} [Can. J. Math. 46, 239-252 (1994; Zbl 0809.05018)] in almost completely determining the spectrum of sets of three pairwise orthogonal Steiner triple systems. In particular they show that such 3OSTS(\(v\)) exist for all \(v \equiv 1 \pmod{6}\) with \(v \geq 19\) (they do not exist for \(v = 7, 13\)), and for all \(v \equiv 3 \pmod{6}\) with \(v \geq 27\), with only \(24\) possible exceptions, the smallest being \(v = 21\). As noted by the authors, a well-followed path is taken, in which they first construct 3OSTS(\(v\)) for many small values of \(v\), and then use Wilson fundamental construction-type recursion to get all of the large orders. The main ingredient in the recursive construction is the orthogonal group divisible design (OGDD), which the authors naturally extend to a set of three pairwise OGDD, or 3OGDD. The direct constructions of many 3OSTS and 3OGDD of small orders were achieved by the use of hill-climbing (as described by \textit{P. B. Gibbons} and \textit{R. Mathon} [J. Comb. Des. 1, 27-50 (1993; Zbl 0817.05014)]) under the assumption of a prescribed automorphism group structure of the target design. As discovered by Colbourn et al., working out the right group structure is the key to the success of the search. For the record, the remaining unknown cases are of orders \(6n+3\) for \(n \in \{3,17,27,29,31,34,35,37,38,40,\dots 51, 57, 58, 59\}\). The authors have little doubt that 3OSTS exist for all \(v \geq 27\), but are not willing to conjecture whether or not a 3OSTS(21) exists. Now there's a challenge!