Existence Of Incomplete Group Divisible Designs

Incomplete group divisible designs (IGDDs) are the group divisible designs (GDDs) missing disjoint sub-GDDs, which need not exist. We denote by \(\text{IGDD}_\text{u}^\text{k}(v, n)\) the design \(\text{GDD}[k, 1, v; uv]\) missing a sub-\(\text{GDD}[k, 1, n; un]\). In this paper, we give the necessary condition for the existence of \(\text{IGDD}_\text{u}^\text{k}(v, n)\) and prove that the necessary condition is also sufficient for \(k = 3\).