Existence of large sets of disjoint group‐divisible designs with block size three and type 2 n 4 1

Large sets of disjoint group‐divisible designs with block size three and type 2 n 4 1 (denoted by LS  (2 n 4 1 )) were first studied by Schellenberg and Stinson and motivated by their connection with perfect threshold schemes. It is known that such large sets can exist only for n  ≡ 0 (mod 3) and do exist for any n  ∉  {12, 36, 48, 144} ∪ {m > 6 : m  ≡  6,30 (mod 36)}. In this paper, we show that an LS  (2 12 k  + 6 4 1 ) exists for any k  ≠ 2. So, the existence of LS  (2 n 4 1 ) is almost solved with five possible exceptions n ∈ {12, 30, 36, 48, 144}. This solution is based on the known existence results of S  (3, 4, v )s by Hanani and special S  (3, {4, 6}, 6 m )s by Mills. Partitionable H ( q , 2, 3, 3) frames also play an important role together with a special known LS  (2 18 4 1 ) with a subdesign LS  (2 6 4 1 ). © 2004 Wiley Periodicals, Inc.