Existence of large sets of disjoint group-divisible designs with block size three and type 2~n4~1

摘要

Large sets of disjoint group-divisible designs with block size three and type 2n41 (denoted by LS (2n41)) 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~(12k+6)4~1) exists for any k ≠ 2. So, the existence of LS (2n41) 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}, 6m)s by Mills. Partitionable H(q, 2, 3, 3) frames also play an important role together with a special known LS (21841) with a subdesign LS (2~64~1).