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Group Divisible Designs with Block Size Four and Group Type g u m 1 for g ≡ 0 mod 6
Author(s) -
Wei Hengjia,
Ge Gennian
Publication year - 2014
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.21336
Subject(s) - mathematics , type (biology) , group (periodic table) , block size , combinatorics , block (permutation group theory) , arithmetic , discrete mathematics , computer science , biology , key (lock) , ecology , computer security , chemistry , organic chemistry
Nonuniform group divisible designs (GDDs) have been studied by numerous researchers for the past two decades due to their essential role in the constructions for other types of designs. In this paper, we investigate the existence problem of { 4 } ‐GDDs of typeg u m 1for g ≡ 0 mod 6 . First, we determine completely the spectrum of { 4 } ‐GDDs of types18 u m 1and36 u m 1 . Furthermore, for general cases, we show that for each g ≡ 0 mod 6 and g ≥ 12 , a { 4 } ‐GDD of typeg u m 1exists if and only if u ≥ 4 , m ≡ 0 mod 3 and 0 ≤ m ≤ g ( u − 1 ) / 2 , except possibly for g ≡ 6 , 42 , 66 , 78 , 102 , 114 , 138 , 174 mod 180 , g ≥ 42 and ( u , m ) ∈ { ( 7 , 3 g − 3 ) , ( 11 , 5 g − 9 ) , ( 11 , 5 g − 6 ) , ( 11 , 5 g − 3 ) , ( 13 , 6 g − 9 ) , ( 13 , 6 g − 3 ) } .