Orthogonal 3-GDDs with four groups

Consider a pair of group divisible designs (GDD) with block size 3, index λ = 1, and on the same points and groups. They are said to be orthogonal (OGDD) if (i) whenever two blocks, one from each design, intersect in two points, the third points are in different groups; and (ii) two disjoint pairs of points defining intersecting triples in one GDD fail to do so in the other. A question posed by Colbourn and Gibbons, in New Zealand J. Math. 7 (1999), 431–440, asks whether there exists any OGDD with four groups. Despite some nonexistence results, their question is answered in the affirmative here with two computer constructions. Also, two algebraic constructions of related structures help toward some asymptotic existence results.