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Existence of large sets of disjoint group‐divisible designs with block size three and type 2 n 4 1
Author(s) -
Ji L.
Publication year - 2005
Publication title -
journal of combinatorial designs
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.618
H-Index - 34
eISSN - 1520-6610
pISSN - 1063-8539
DOI - 10.1002/jcd.20039
Subject(s) - combinatorics , mathematics , disjoint sets , type (biology) , block size , group (periodic table) , block (permutation group theory) , connection (principal bundle) , discrete mathematics , arithmetic , physics , computer science , ecology , geometry , computer security , key (lock) , biology , quantum mechanics
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.