Embeddings and faithful enclosings of GDDs with a constant number of groups

We introduce, for block size 3, the problem of minimal enclosings of GDDs with u groups of size g into GDDs with u groups of size g + 1. We prove the necessary conditions are sufficient for the embedding of any GDD with group size 1, i.e., a BIBD(v, 3, λ), into a GDD with v groups of size 2. Minimal faithful enclosings of GDDs are given for group size 2 with any index into GDDs with group size 3, and we give some non-minimal faithful enclosings for group size 2 and any index. The minimality in several cases is shown by applying a new necessary condition for enclosing of GDDs.