Group Divisible Designs Research Articles

Formula omitted]-cyclic holey group divisible designs and their application to three-dimensional optical orthogonal codes

Motivated by the application to three-dimensional optical orthogonal codes, we consider the constructions for a w-cyclic holey group divisible design of type (u,wv) with block size k, which is denoted by w-cyclic k-HGDD of type (u,wv). The necessary conditions of such a design, namely u,v≥k, (u−1)(v−1)w≡0(modk−1) and u(u−1)v(v−1)w≡0(modk(k−1)), are shown to be sufficient for k=3. As an application, we give a complete solution of the existence problem of perfect three-dimensional optical orthogonal codes of weight three with both at most one-pulse per spatial plane and at most one-pulse per wavelength plane properties.

Resolvable Group Divisible Designs with Large Groups

We prove that the necessary divisibility conditions are sufficient for the existence of resolvable group divisible designs with a fixed number of sufficiently large groups. Our method combines an application of the Rees product construction with a streamlined recursion based on incomplete transversal designs. With similar techniques, we also obtain new results on decompositions of complete multipartite graphs into a prescribed graph.

Determination of Sizes of Optimal Three‐Dimensional Optical Orthogonal Codes of Weight Three with the AM‐OPP Restriction

AbstractIn this paper, we further investigate the constructions on three‐dimensional optical orthogonal codes with the at most one optical pulse per wavelength/time plane restriction (briefly AM‐OPP 3D ‐OOCs) by way of the corresponding designs. Several new auxiliary designs such as incomplete holey group divisible designs and incomplete group divisible packings are introduced and therefore new constructions are presented. As a consequence, the exact number of codewords of an optimal AM‐OPP 3D ‐OOC is finally determined for any positive integers and .

Group divisible designs of four groups and block size five with configuration (1; 1; 1; 2)

We present constructions and results about GDDs with four groups and block size five in which each block has Configuration $(1, 1, 1, 2)$, that is, each block has exactly one point from three of the four groups and two points from the fourth group. We provide the necessary conditions of the existence of a GDD$(n, 4, 5; \lambda_1, \lambda_2)$ with Configuration $(1, 1, 1, 2)$, and show that the necessary conditions are sufficient for a GDD$(n, 4, 5; \lambda_1,$ $\lambda_2)$ with Configuration $(1, 1, 1, 2)$ if $n \not \equiv 0 ($mod $6)$, respectively. We also show that a GDD$(n, 4, 5; 2n, 6(n - 1))$ with Configuration $(1, 1, 1, 2)$ exists, and provide constructions for a GDD$(n = 2t, 4, 5; n, 3(n - 1))$ with Configuration $(1, 1, 1, 2)$ where $n \not= 12$, and a GDD$(n = 6t, 4, 5; 4t, 2(6t - 1))$ with Configuration $(1, 1, 1, 2)$ where $n \not= 6$ and $18$, respectively.

2-Walk-regular graphs with a small number of vertices compared to the valency

In 2013, it was shown that, for a given real number α>2, there are only finitely many distance-regular graphs Γ with valency k and diameter D≥3 having at most αk vertices, except for the following two cases: (i) D=3 and Γ is imprimitive; (ii)D=4 and Γ is antipodal and bipartite. In this paper, we will generalize this result to 2-walk-regular graphs. In this case, also incidence graphs of certain group divisible designs appear.

2-Walk-Regular Dihedrants from Group-Divisible Designs

In this note, we construct bipartite $2$-walk-regular graphs with exactly 6 distinct eigenvalues as the point-block incidence graphs of group divisible designs with the dual property. For many of them, we show that they are 2-arc-transitive dihedrants. We note that some of these graphs are not described in Du et al. (2008), in which they classified the connected 2-arc transitive dihedrants.

Characterization of Group Divisible Designs

In this paper, some inequalities of PBIB designs have been given. In this paper, the characterization of all the three types of GD designs, by finding some equalities and inequalities among the parameters of GD designs is obtained.

A New Construction of Group Divisible Designs with Nonuniform Group Type

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 gkm1 and block size k. We make a progress by proposing a new construction, in which generalized difference sets, a truncating technique, and a difference method are combined to construct nonuniform GDDs. Moreover, we present a variation of this new construction by employing Rees' product constructions. We obtain several infinite families of nonuniform GDDs, as well as many examples whose block sizes are relatively large.

Matching divisible designs with block size four

We consider edge-decompositions of the graph join of several equal-sized one-factors into cliques of a prescribed size. These objects are variants of group divisible designs and have applications to packings, coverings, and embeddings. Assuming block (clique) size four, we show that the obvious divisibility and counting conditions are sufficient for the existence of such designs.

HFR code: a flexible replication scheme for cloud storage systems

Fractional repetition (FR) codes are a family of repair-efficient storage codes that provide exact and uncoded node repair at the minimum bandwidth regenerating point. The advantageous repair properties are achieved by a tailor-made two-layer encoding scheme which concatenates an outer maximum-distance-separable (MDS) code and an inner repetition code. In this paper, we generalize the application of FR codes and propose heterogeneous fractional repetition (HFR) code, which is adaptable to the scenario where the repetition degrees of coded packets are different. We provide explicit code constructions by utilizing group divisible designs, which allow the design of HFR codes over a large range of parameters. The constructed codes achieve the system storage capacity under random access repair and have multiple repair alternatives for node failures. Further, we take advantage of the systematic feature of MDS codes and present a novel design framework of HFR codes, in which storage nodes can be wisely partitioned into clusters such that data reconstruction time can be reduced when contacting nodes in the same cluster.

The Existence of Well‐Balanced Triple Systems

AbstractA triple system is a collection of b blocks, each of size three, on a set of v points. It is j‐balanced when every two j‐sets of points appear in numbers of blocks that are as nearly equal as possible, and well balanced when it is j‐balanced for each . Well‐balanced systems arise in the minimization of variance in file availability in distributed file systems. It is shown that when a triple system that is 2‐balanced and 3‐balanced exists, so does one that is well balanced. Using known and new results on variants of group divisible designs, constructions for well‐balanced triple systems are developed. Using these, the spectrum of pairs for which such a well‐balanced triple system exists is determined completely.

Some new resolvable GDDs with [formula omitted] and doubly resolvable GDDs with [formula omitted

A doubly resolvable packing design with block size k, index λ, replication number r, and v elements is called a generalized Kirkman square and denoted by GKSk(v;1,λ;r). Existence of GKS3(4u;1,1;2(u−1))s and GKS3(6u;1,1;3(u−1))s is implied by existence of doubly resolvable group divisible designs with block size 3, index 1, and types 4u and 6u (i.e., (3,1)-DRGDDs of types 4u and 6u). In this paper, we establish the spectra of (3,1)-DRGDDs of types 4u and 6u with 15 and 31 possible exceptions, respectively. As applications, we get some new classes of permutation codes and doubly constant weight codes. We also construct 5 new resolvable GDDs with block size 4 and index 1.

The Use of Treatment Concurrences to Assess Robustness of Binary Block Designs Against the Loss of Whole Blocks

SummaryCriteria are proposed for assessing the robustness of a binary block design against the loss of whole blocks, based on summing entries of selected upper non‐principal sections of the concurrence matrix. These criteria improve on the minimal concurrence concept that has been used previously and provide new conditions for measuring the robustness status of a design. The robustness properties of two‐associate partially balanced designs are considered and it is shown that two categories of group divisible designs are maximally robust. These results expand a classic result in the literature, obtained by Ghosh, which established maximal robustness for the class of balanced block designs.

Constructions of large sets of disjoint group-divisible designs [formula omitted] using a generalization of [formula omitted

Large sets of disjoint group-divisible designs with block size three and type 2n41 were first studied by Schellenberg and Stinson and motivated by their connection with perfect threshold schemes. It is known that such large sets can exist only for n≡0(mod3) and do exist for n=2k(3m), where m≡1(mod2) and k=0,3 or k≥5. A special large set called *LS(2n) has played a key role in obtaining the above results. In this paper, we shall give a generalization of an *LS(2n) and use it to obtain a similar result for k=2,4 and partially for k=1.

Graph Divisible Designs and Packing Constructions

We introduce a generalization of group divisible designs and offer example applications to challenging problems in design theory. The generalization considers edge-decompositions of joins of arbitrary graphs, whereas group divisible designs handle only joins of edgeless graphs. Our example constructions include: (1) optimal packings with block size five for the previously unsettled congruence class $$v \equiv 13 \pmod {20}$$v?13(mod20); (2) an optimal grooming with with ratio seven for the previously unsettled congruence class $$v \equiv 56 \pmod {84}$$v?56(mod84); and (3) a constructive `quadratic' embedding of partial designs with block size four.

三个组的<em>G-</em>设计

G-设计是可分组设计(GD)的推广，同时又是烛台型设计(CQS)的特例，它在四元系设计中起到重要作用。文章应用Stern和Lenz关于图因子分解的结论，通过直接构造法，得到具有三个组的G-设计存在的充分必要条件：。 As a special example of the candelabra systems (CQS), G-design is the extension of group divisible designs (GD), which plays an important role in quadruple systems’ construction. With application of Stern and Lenz’s result on one-factorization of graphs, by direct construction, it is given that the sufficient and necessary condition for the existence of the G-design with three groups is that.

Semi-cyclic Holey Group Divisible Designs and Applications to Sampling Designs and Optical Orthogonal Codes

We consider the existence problem for a semi-cyclic holey group divisible design of type with block size 3, which is denoted by a 3-SCHGDD of type . When t is odd and or t is doubly even and , the existence problem is completely solved; when t is singly even, many infinite families are obtained. Applications of our results to two-dimensional balanced sampling plans and optimal two-dimensional optical orthogonal codes are also discussed.

Super-simple (5, 4)-GDDs of group type g u

In statistical planning of experiments, super-simple designs are the ones providing samples with maximum intersection as small as possible. Supersimple group divisible designs are useful in constructing other types of supersimple designs which can be applied to codes and designs. In this article, the existence of a super-simple (5, 4)-GDD of group type gu is investigated and it is shown that such a design exists if and only if u ⩾ 5, g(u − 2) ⩾ 12, and u(u − 1)g2 ≡ 0 (mod 5) with some possible exceptions.

Group divisible designs with block sizes from [formula omitted] and Kirkman frames of type [formula omitted

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 K1(3)-GDDs of type gum1 with K1(3)={k:k≡1mod3} and Kirkman frames of type hum1. First, we determine completely the spectrum for uniform K1(3)-GDDs of type gu. Then, we consider the entire existence problem for non-uniform K1(3)-GDDs of type gum1 with m>0. We show that, for each given g, up to a small number of undetermined cases of u, the necessary conditions on (u,m) for the existence of a K1(3)-GDD of type gum1 are also sufficient, except possibly when u≡2mod4 for g≡3mod6 and when u≡6mod12 for g≡1,5mod6. Finally, a similar result for Kirkman frames of type hum1 is obtained. We show that, for each given h, up to a small number of undetermined cases of u, the necessary conditions on (u,m) for the existence of a Kirkman frame of type hum1 are also sufficient, except possibly when u≡2mod4 for h≡6mod12 and when u≡6mod12 for h≡2,10mod12.

General Fractional Repetition Codes for Distributed Storage Systems

In order to provide fault-tolerance and guarantee reliability, data redundancy should be introduced in distributed storage systems. The emerging coding techniques for storage such as fractional repetition codes, provide the required redundancy more efficiently than the conventional replication scheme. In this letter, we extend the construction of fractional repetition codes and present a new coding scheme, termed general fractional repetition codes. The proposed codes can be applied to storage systems in which the storage capacities of nodes may be different. Based on a combinatorial structure known as group divisible design, the new code construction is available for a large set of parameters. The performance of the proposed codes is evaluated by a metric called node repair alternativity, which measures the number of different subsets of nodes that enable the repair of a specific failed node.