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Abstract: 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 ≡ 7 mod 12 and u ≡ 3 mod 6, when g ≡ 11 , 14 , 17 , 23 mod 24 and u ≡ 5 mod 6, and in several small families for which one of g and u is fixed. [ABSTRACT FROM AUTHOR]

The research on directed PBDs is motivated by the construction of t-deletion/insertion-correcting codes. Fuji-Hara, Miao, Wang, and Yin have determined the existence of directed PBDs with block sizes from the set [ABSTRACT FROM AUTHOR]

The construction of group divisible designs (GDDs) is a basic problem in design theory. While there have been some methods concerning the constructions of uniform GDDs, the construction of nonuniform GDDs remains a challenging problem. In this paper, we present a new approach to the construction of nonuniform GDDs with group type [ABSTRACT FROM AUTHOR]

In this paper, we continue to investigate the existence of 5 -GDDs, 4-frames and 4-RGDDs, which have been studied by numerous researchers in the past two decades due to their vital applications in the constructions for other types of designs. Much progress has been made for the existence problems of these designs, while the problems are still open. Our new results mainly lie in the cases with relatively large group size. We also construct some smaller designs, including 4 -RGDDs of types 2 184 , 9 44 , 14 10 , 18 18 , 22 10 and 36 11 . [ABSTRACT FROM AUTHOR]

Abstract: Non-uniform group divisible designs (GDDs) and non-uniform Kirkman frames are useful in the constructions for other types of designs. In this paper, we consider the existence problems for -GDDs of type with and Kirkman frames of type . First, we determine completely the spectrum for uniform -GDDs of type . Then, we consider the entire existence problem for non-uniform -GDDs of type with . We show that, for each given , up to a small number of undetermined cases of , the necessary conditions on for the existence of a -GDD of type are also sufficient, except possibly when for and when for . Finally, a similar result for Kirkman frames of type is obtained. We show that, for each given , up to a small number of undetermined cases of , the necessary conditions on for the existence of a Kirkman frame of type are also sufficient, except possibly when for and when for . [Copyright &y& Elsevier]

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 [ABSTRACT FROM AUTHOR]

Abstract: Non-uniform group divisible designs are instrumental in the constructions for other types of designs. Most of the progress for the existence of -GDDs of type is on the case when is even, where the existence for small has played a key role. In order to determine the spectrum for -GDDs of type with being odd, we continue to investigate the small cases with in this paper. We show that, for each , the necessary conditions for the existence of a -GDD of type are also sufficient. As the applications of these GDDs, we obtain a few pairwise balanced designs with minimum block size 4. Meanwhile, we also improve the existence result for frame self-orthogonal Mendelsohn triple systems of type by reducing an infinite class of possible exceptions, namely and , to eight undetermined cases. [Copyright &y& Elsevier]

Abstract: In this paper, we investigate the existence of resolvable group divisible designs (RGDDs) with block size four, group-type and general index . The necessary conditions for the existence of such a design are , and . These necessary conditions are shown to be sufficient for all , with the definite exceptions of . The known existence result for is also improved. [Copyright &y& Elsevier]

Abstract: In this paper, we investigate the existence of resolvable group divisible designs (RGDDs) with block size four, group-type and index three. The necessary conditions for the existence of such a design are and . These necessary conditions are shown to be sufficient except for and possibly excepting . [Copyright &y& Elsevier]

In this paper, we present a systematic approach to some new direct constructions of block designs. Several examples of non-uniform group divisible designs, resolvable group divisible designs and optimal packings are illustrated to show the advantage of the construction method. [Copyright &y& Elsevier]

In this paper, we continue to investigate the spectrum for {4}-GDDs of type gum1 with small g. We show that, for each g∈{2,4,5,12,15} and each admissible pair (u,m), a {4}-GDD of type gum1 exists with a definite exception for (g,u,m)=(2,6,5) and 5 possible exceptions when g=2. [Copyright &y& Elsevier]

We determine, up to 13 possible exceptions, the spectrum for {4}-GDDs with group-type 6um1. As an application, we construct a new class of Kirkman frames and use these to complete the spectrum for incomplete nearly Kirkman triple systems INKTS(v,w) where v−w≡0 mod 12. [Copyright &y& Elsevier]

In this paper, a survey for the existence of resolvable group divisible designs (RGDDs) with block size four, group-type hn and index unity is presented. It is proved that there exists a 4-RGDD of type hn if and only if n⩾4, hn≡0 (mod 4) and h(n−1)≡0 (mod 3) with a handful of possible exceptions. [Copyright &y& Elsevier]

A mandatory representation design MRD(K;v) is a pairwise balanced design on v points with block sizes from the set K in which for each k∈K there is at least one block in the design of size k. It was shown by Gru¨ttmu¨ller and Rees (Utilitas Math. 60 (2001) 153–180.) that MRDs with K={4,k} and k≡1 mod 3 exist with at most 56 possible exceptions. In this paper, we continue to investigate the spectrum for such MRDs. It is shown that the necessary conditions for existence are sufficient, with at most 10 possible exceptions. As a preliminary result, some new 4-GDDs of type gum1 are also constructed. [Copyright &y& Elsevier]