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Group divisible covering designs with block size four
Author(s) -
Wei Hengjia,
Ge Gennian,
Colbourn Charles J.
Publication year - 2018
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.21596
Subject(s) - mathematics , combinatorics , generalization , group (periodic table) , block (permutation group theory) , type (biology) , block size , arithmetic , discrete mathematics , computer science , mathematical analysis , ecology , chemistry , computer security , organic chemistry , key (lock) , biology
Group divisible covering designs (GDCDs) were introduced by Heinrich and Yin as a natural generalization of both covering designs and group divisible designs. They have applications in software testing and universal data compression. The minimum number of blocks in a k ‐GDCD of type g u is a covering number denoted by C ( k , g u ) . When k = 3 , the values of C ( 3 , g u ) have been determined completely for all possible pairs ( g , u ) . When k = 4 , Francetić et al. constructed many families of optimal GDCDs, but the determination remained far from complete. In this paper, two specific 4‐IGDDs are constructed, thereby completing the existence problem for 4‐IGDDs of type( g , h ) u . Then, additional families of optimal 4‐GDCDs are constructed. Consequently the cases for ( g , u ) whose status remains undetermined arise when g ≡ 7mod 12 and u ≡ 3mod 6 , when g ≡ 11 , 14 , 17 , 23mod 24 and u ≡ 5mod 6 , and in several small families for which one of g and u is fixed.