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Semiframes with Block Size Three and Odd Group Size
Author(s) -
Cao H.,
Fan J.,
Xu D.
Publication year - 2015
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.21407
Subject(s) - mathematics , partition (number theory) , combinatorics , disjoint sets , block (permutation group theory) , block size , type (biology) , class (philosophy) , group (periodic table) , discrete mathematics , computer science , ecology , chemistry , computer security , organic chemistry , artificial intelligence , key (lock) , biology
A ( k , λ ) ‐semiframe of type g u is a ( k , λ ) ‐GDD of type g u , ( X , G , B ) , in which the collection of blocks B can be written as a disjoint union B = P ∪ Q where Q is partitioned into parallel classes of X and P is partitioned into holey parallel classes, each holey parallel class being a partition of X∖ G jfor someG j ∈ G . A ( k , λ ) ‐SF ( p , d , g u ) is a ( k , λ ) ‐semiframe of type g u in which there are p parallel classes in Q and d holey parallel classes with respect toG j ∈ G . In this paper, we shall show that there exists a (3, 1)‐SF ( p , d , g u ) for any g ≡ 3 ( mod 6 ) if and only if u ≥ 5 , u ≡ 1 ( mod 2 ) , p ≡ u − 1 2( mod u − 1 ) , d = g 2 − p u − 1 , and ( p , d , g , u ) ≠ ( 2 , 1 , 3 , 5 ) .