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Some new group divisible designs with block size 4 and two or three group sizes
Author(s) -
Abel R. Julian R.,
Bunjamin Yudhistira A.,
Combe Diana
Publication year - 2020
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.21719
Subject(s) - block size , mathematics , block (permutation group theory) , combinatorics , group (periodic table) , type (biology) , discrete mathematics , computer science , biology , physics , key (lock) , ecology , computer security , quantum mechanics
Abstract Group divisible designs (GDDs) with block size 4 and at most 30 points are known for all feasible group types except three, namely 2 3 5 4 , 3 5 6 2 , and 2 2 5 5 . In this paper we provide solutions for the first two of these three 4‐GDDs without assuming any automorphisms. We also construct several other 4‐GDDs. These include classes of 4‐GDDs of types( 3 m ) 4( 6 m ) q( 3 n ) 1 for 0 ≤ n ≤ ( q + 1 ) m where q ∈ { 2 , 3 } and solutions for 4‐GDDs of types 3 t 6 s for a wide range of values of s ≥ 2 , t ≥ 4 satisfying t ≡ 0 or 1 ( mod 4 ) , including all cases with 4 ≤ t ≤ s − 1 . Most of the remaining unknown 4‐GDDs of type 3 t 6 s have ( s − 1 ) < t < 2 ( s − 1 ) .