Group Divisible Designs Research Articles

An Improvement on Triple Systems Without Two Types of Configurations

ABSTRACTThere are four nonisomorphic configurations of triples that can form a triangle in a three‐uniform hypergraph, where the configurations and on consist of three triples and , respectively. Denote by ex and ex the maximum number of triples in a three‐uniform hypergraph on vertices which does not contain , both and , respectively. Recently, Frankl et al. used theorem of Gustavsson on sufficiently dense graphs to determine ex and ex for all . In this note, we use packings and group divisible designs of block size 4 to remove the condition .

Key Pre-distribution Schemes via Certain Resolvable Block Designs

Earlier optimal key pre\(-\)distribution schemes (KPSs) for distributed sensor networks (DSNs) were proposed using combinatorial designs via transversal designs, affine, and partially affine resolvable designs. Here, nearly optimal KPSs are introduced and a class of such KPSs is obtained from resolvable group divisible designs. These KPSs are nearly optimal in the sense of local connectivity. A metric for the efficiency of KPSs is given. Further, an optimal KPS has also been proposed using affine resolvable \( L_{2} \)-type design.

A Survey on Partially Resolvable Solutions of Regular Group Divisible Designs

A Survey on Partially Resolvable Solutions of Regular Group Divisible Designs

On Constructions of GMGDs

Modified group divisible designs MGD ( k , λ , m , n ) are extensively studied because of an intriguing combinatorial structure that they possess and their applications. In this paper, we present a generalization of MGDs called GMGD ( k , λ 1 , λ 2 , m , n ) , and we provide some elementary results and constructions of some special cases of GMGDs. In addition, we show that the necessary conditions are sufficient for the existence of a GMGD ( 3 , λ , 2 λ , m , n ) for any positive integer λ , and a GMGD ( 3 , 2 , 3 , m , n ) . Though not a general result, the construction of a GMGD ( 3 , 3 , 2 , 2 , 6 ) given in the paper is worth mentioning in the abstract. Along with another example of a GMGD ( 3 , 3 , 2 , 2 , 4 ) , and n to t n construction, we have families of GMGD ( 3 , 3 λ , 2 λ , 2 , n ) s for n = 4 t or 6 t when t ≡ 0 , 1 ( mod 3 ) , for any positive integer λ .

Group divisible designs with block size 4 and group sizes 4 and 7

AbstractIn this paper, we consider the existence of group divisible designs (GDDs) with block size and group sizes and . We show that there exists a ‐GDD of type for all but a finite specified set of feasible values for .

Block Structured Hadamard matrices from certain arrays

We have constructed Block structured Hadamard matrices in which odd number of blocks are used in a row (column). These matrices are different than those introduced by Agaian. Generalised forms of arrays developed by Goethals-Seidel, Wallis-Whiteman and Seberry-Balonin heve been employed. Such types of matrices are applicable in the constructions of nested group divisible designs.

3-Group Divisible Designs with 3 Groups and Block Size 5

A 3 -GDD ( n , 2 , k , λ 1 , λ 2 ) was defined by combining the definitions of a group divisible design and a t-design. In this paper, we extend the definitions to 3 groups and block size 5, and we denote such GDD by 3 -GDD ( n , 3 , 5 , μ 1 , μ 2 ). Some necessary conditions for the existence of these GDDs are developed, and several new constructions and specific instances of nonexistence are given.

Frames and Doubly Resolvable Group Divisible Designs with Block Size Three and Index Two

Frames and Doubly Resolvable Group Divisible Designs with Block Size Three and Index Two

A generalization of group divisible t $t$‐designs

AbstractCameron defined the concept of generalized ‐designs, which generalized ‐designs, resolvable designs and orthogonal arrays. This paper introduces a new class of combinatorial designs which simultaneously provide a generalization of both generalized ‐designs and group divisible ‐designs. In certain cases, we derive necessary conditions for the existence of generalized group divisible ‐designs, and then point out close connections with various well‐known classes of designs, including mixed orthogonal arrays, factorizations of the complete multipartite graphs, large sets of group divisible designs, and group divisible designs with (orthogonal) resolvability. Moreover, we investigate constructions for generalized group divisible ‐designs and almost completely determine their existence for and small block sizes.

On the existence of k $k$‐cycle semiframes for even k $k$

AbstractA ‐semiframe of type is a ‐group divisible design of type in which is the vertex set, is the group set, and the set of ‐cycles can be written as a disjoint union where is partitioned into parallel classes on and is partitioned into holey parallel classes, each parallel class or holey parallel class being a set of vertex disjoint cycles whose vertex sets partition or for some . In this paper, we almost completely solve the existence of a ‐semiframe of type for all and a ‐semiframe of type for all and .

Optimal key pre‐distribution schemes from affine resolvable and partially affine resolvable designs

AbstractVarious key pre‐distribution schemes (KPSs) for distributed sensor networks (DSNs) were obtained from different combinatorial designs. A block design may be mapped to a KPS where is the number of nodes or network size and is the number of keys (points) per node. A KPS is corresponding to a symmetric balanced incomplete block design has full connectivity where all nodes are connected to each other. But it is desirable that all nodes do not share common keys as the compromise of any node will destroy the whole network (see Lee & Stinson, 2005; Martin, 2009). To meet the situation, Lee and Stinson (2005) introduced μ—common intersection design which has disjoint blocks and these disjoint blocks can establish a secure communication through certain common intersecting blocks. Here we have proposed optimal KPSs from affine resolvable and partially affine resolvable balanced incomplete block designs, group divisible designs and L2—type designs. It is shown that these designs are important classes of μ—common intersection designs. The KPSs obtained here are optimal in the sense of local connectivity. Mathematics Subject Classification (2010): 94C30; 68R05; 05B05; 62K10.

Algebraic constructions of group divisible designs

Some series of Group divisible designs using generalized Bhaskar Rao designs over Dihedral, Symmetric and Alternating groups are obtained.

Group divisible designs with block size 4 where the group sizes are congruent to 1 mod 3

In this paper we consider the existence of 4-GDDs on v points with v≤50. We consider the case v≡1(mod3) and construct designs for all feasible group types here. We also enumerate all 4-GDDs with type 74 and compare the actions of their automorphism groups. When v≡0(mod3) and v≤50, there are no feasible group types for which existence has not been determined, but for v≡2(mod3) and v≤50, we obtain 4-GDDs for two more feasible group types.

Balanced Arrays Through Symmetrical Regular Group Divisible Designs

An alternative and simple method of construction of balanced array, orthogonal and proportional frequency main effect plans and asymmetrical non-orthogonal main effect plans using incidence matrix of symmetrical regular GD designs has been proposed. The above plans can be constructed by symmetrical regular GD designs only when p is a positive integer.

Perfect secret sharing schemes from combinatorial squares

AbstractChaudhry et al. proposed perfect secret sharing schemes from combinatorial squares called Room squares based on certain balanced incomplete block designs. Their protocol is efficient and secure but it is based on a Room square of order r which exists if and only if r is odd and r ≠ 3, 5. Here, we have proposed the schemes from Room squares based on group divisible designs which are a broader class of designs.

Constructions of Sarvate-Beam Group Divisible Designs

A balanced incomplete block design is a set system in which all pairs of distinct elements occur with a constant frequency. By contrast, a Sarvate-Beam design induces an interval of distinct frequencies on pairs. In this paper, we settle the existence of a Sarvate-Beam variant of group divisible designs of uniform type with block size three.

Generalized perfect difference families and their application to variable-weight geometric orthogonal codes

Motivated by the application in geometric orthogonal codes (GOCs), Wang et al. introduced the concept of generalized perfect difference families (PDFs), and established the equivalence between GOCs and a certain type of generalized PDFs recently. Based on the relationship, we discuss the existence problem of generalized (n×m,K,1)-PDFs in this paper. By using some auxiliary designs such as semi-perfect group divisible designs and several recursive constructions, we prove that a generalized (n×m,{3,4},1)-PDF exists if and only if nm≡1(mod6). The existence of a generalized (n×m,{3,4,5},1)-PDF is also completely solved possibly except for a few values. As a consequence, some variable-weight perfect (n×m,K,1)-GOCs are obtained.

An update on the existence of Kirkman triple systems with steiner triple systems as subdesigns

AbstractA Kirkman triple system of order , KTS, is a resolvable Steiner triple system on elements. In this paper, we investigate an open problem posed by Doug Stinson, namely the existence of KTS which contain as a subdesign a Steiner triple system of order , an STS. We present several different constructions for designs of this form. As a consequence, we completely settle the extremal case , for which a list of possible exceptions had remained for close to 30 years. Our new constructions also provide the first infinite classes for the more general problem. We reduce the other maximal case to (at present) three possible exceptions. In addition, we obtain results for other cases of the form and also near . Our primary method introduces a new type of Kirkman frame which contains group divisible design subsystems. These subsystems can occur with different configurations, and we use two different varieties in our constructions.

A note on $GDD(1, n, n , 4;\lambda_{1},\lambda_{2})$

The present note is motivated by two papers on group divisible designs (GDDs) with the same block size three but different number of groups: three and four where one group is of size $1$ and the others are of the same size $n$. Here we present some interesting constructions of GDDs with block size 4 and three groups: one of size $1$ and other two of the same size $n$. We also obtain necessary conditions for the existence of such GDDs and prove that they are sufficient in several cases. For example, we show that the necessary conditions are sufficient for the existence of a GDD$(1,n,n,4;\lambda_1,\lambda_2)$ for $n\equiv0,1,4,5,8,9\pmod{12}$ when $\lambda_1\ge \lambda_2$.

Group divisible designs with block size 4 and group sizes 2 and 5

AbstractIn this paper we provide a 4‐GDD of type , thereby solving the existence question for the last remaining feasible type for a 4‐GDD with no more than 30 points. We then show that 4‐GDDs of type exist for all but a finite specified set of feasible pairs .