TY - JOUR
T1 - Group divisible covering designs with block size four
AU - Wei, Hengjia
AU - Ge, Gennian
AU - Colbourn, Charles
N1 - Funding Information:
Gennian Ge, School of Mathematical Sciences, Capital Normal University, Beijing 100048, China. Email: gnge@zju.edu.cn Grant sponsor: Post-Doctoral Science Foundation of China; Grant number: 2015M571067; Grant sponsor: Beijing Postdoctoral Research Foundation; Grant sponsor: National Natural Science Foundation of China; Grant numbers: 11431003 and 61571310; Grant sponsor: Beijing Scholars Program; Grant sponsor: Beijing Hundreds of Leading Talents Training Project of Science and Technology; Grant sponsor: Beijing Municipal Natural Science Foundation; Grant sponsor: National Science Foundation; Grant number 1421058.
Publisher Copyright:
© 2017 Wiley Periodicals, Inc.
PY - 2018/3
Y1 - 2018/3
N2 - 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 gu is a covering number denoted by C(k, gu). When k = 3, the values of C(3, gu) 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 ≡ 11, 14, 17, 23 and 24, when u ≡ 5 and 6, and in several small families for which one of g and u is fixed.
AB - 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 gu is a covering number denoted by C(k, gu). When k = 3, the values of C(3, gu) 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 ≡ 11, 14, 17, 23 and 24, when u ≡ 5 and 6, and in several small families for which one of g and u is fixed.
KW - covering numbers
KW - group divisible covering designs
KW - group divisible designs
KW - incomplete group divisible designs
KW - primary 05B05
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U2 - 10.1002/jcd.21596
DO - 10.1002/jcd.21596
M3 - Article
AN - SCOPUS:85037613566
SN - 1063-8539
VL - 26
SP - 101
EP - 118
JO - Journal of Combinatorial Designs
JF - Journal of Combinatorial Designs
IS - 3
ER -