A structural classification of SR group divisible designs

A simple method of construction of a semi-regular (SR) group divisible design from another SR group divisible design is given. Using this method, 111 available SR designs from Clatworthy (1973) and John and Turner (1977) are systematically classified into 20 classes. This procedure may produce new nonisomorphic solutions for known designs.     

Resolvable modified group divisible designs with block size three

A resolvable modified group divisible design (RMGDD) is an MGDD whose blocks can be partitioned into parallel classes. In this article, we investigate the existence of RMGDDs with block size three and show that the necessary conditions are also sufficient with two exceptions. © 2005 Wiley Periodicals, Inc. J Combin Designs 15: 2–14, 2007     

区组大小为 4 的二重超单可分解的可分组设计

自 1992 年 Gronau 和 Mullin 提出超单设计的概念以来, 很多研究者参与了超单设计的研究. 超单设计在编码等方面也有广泛的应用. 超单可分组设计是超单设计的重要组成部分. 本文我们主要研究区组大小为4 的二重超单可分解的可分组设计, 并基本解决了此类设计的存在性问题.     

α‐Resolvable group divisible designs with block size three

A group divisible design GD(k,λ,t;tu) is α‐resolvable if its blocks can be partitioned into classes such that each point of the design occurs in precisely α blocks in each class. The necessary conditions for the existence of such a design are λt(u ? 1) = r(k ? 1), bk = rtu, k|αtu and α|r. It is shown in this paper that these conditions are also sufficient when k = 3, with some definite exceptions. © 2004 Wiley Periodicals, Inc.     

Balanced arrays of strength two from block designs

Some constructions of balanced arrays of strength two are provided by use of rectangular designs, group divisible designs, and nested balanced incomplete block designs. Some series of such arrays are also presented as well as orthogonal arrays, with illustrations. © 2002 Wiley Periodicals, Inc. J Combin Designs 10: 303–312, 2002; Published online in Wiley InterScience ( www.interscience.wiley.com ). DOI 10.1002/jcd.10016     

A construction of resolvable nested 3‐designs

A method of constructing resolvable nested 3‐designs from an affine resolvable 3‐design is proposed with one example. © 2004 Wiley Periodicals, Inc.     

D‐optimal designs and group divisible designs

We obtain new conditions on the existence of a square matrix whose Gram matrix has a block structure with certain properties, including D‐optimal designs of order , and investigate relations to group divisible designs. We also find a matrix with large determinant for n = 39. © 2006 Wiley Periodicals, Inc. J Combin Designs 14: 451–462, 2006     

Embeddability and construction of affine α-resolvable pairwise balanced designs

An affine α-resolvable PBD of index λ is a triple (V, B, R), where V is a set (of points), B is a collection of subsets of V (blocks), and R is a partition of B (resolution), satisfying the following conditions: (i) any two points occur together in λ blocks, (ii) any point occurs in α blocks of each resolution class, and (iii) |B| = |V| + |R| − 1. Those designs embeddable in symmetric designs are described and two infinite series of embeddable designs are constructed. The analog of the Bruck–Ryser–Chowla theorem for affine α-resolvable PBDs is obtained. © 1998 John Wiley & Sons, Inc. J Combin Designs 6:111–129, 1998     

Spreads and group divisible designs

By removing the components of at-spread of a finite projective spacePG(d, q) from each hyperplane ofPG(d, q), the blocks of a regular group divisible design are obtained We characterize geometrict-spreads as thoset-spreads which are such that the dual of is also a group divisible design.     

Embeddings of resolvable group divisible designs with block size 3

It is proved in this paper that an RGD(3, g;v) can be embedded in an RGD(3, g;u) if and only if , , , v ≥ 3g, u ≥ 3v, and (g,v) ≠ (2,6),(2,12),(6,18).     

Incomplete group divisible designs with block size four and general index

In this paper, we investigate the existence of incomplete group divisible designs (IGDDs) with block size four, group-type (g, h) u and general index λ. The necessary conditions for the existence of such a design are that u ≥ 4, g ≥ 3h, λg(u 1) ≡ 0 (mod 3), λ(g h)(u 1) ≡ 0 (mod 3), and λu(u 1)(g 2 h 2 ) ≡ 0 (mod 12). These necessary conditions are shown to be sufficient for all λ≥ 2. The known existence result for λ = 1 is also improved.     

Lifting of divisible designs

The aim of this paper is to present a construction of t-divisible designs (DDs) for t > 3, because such DDs seem to be missing in the literature. To this end, tools such as finite projective spaces and their algebraic varieties are employed. More precisely, in a first step an abstract construction, called t-lifting, is developed. It starts from a set X containing a t-DD and a group G acting on X. Then several explicit examples are given, where X is a subset of PG(n,q) and G is a subgroup of GL_n + 1(q). In some cases X is obtained from a cone with a Veronesean or an h-sphere as its basis. In other examples, X arises from a projective embedding of a Witt design. As a result, for any integer t ≥ 2 infinitely many non-isomorphic t-DDs are found.Dedicated to Walter Benz on the occasion of his 75th birthday.     

Super-simple group divisible designs with block size 4 and index 5

In this paper, we investigate the existence of a super-simple (4, 5)-GDD of type g^u and show that such a design exists if and only if u≥4, g(u−2)≥10, and .     

A construction of group divisible designs with block sizes 3 to 7

This paper gives a construction of group divisible designs (GDDs) on the binary extension fields with block sizes 3, 4, 5, 6, and 7, respectively, which consist of the error patterns whose first syndromes are zeros recognized from the decoding of binary quadratic residue codes. A conjecture is proposed for this construction of GDDs with larger block sizes.     

The spectrum of semicyclic holey group divisible designs with block size three

We determine a necessary and sufficient condition for the existence of semicyclic holey group divisible designs with block size three and group type $\left(n,m^t\right)$. New recursive constructions on semicyclic incomplete holey group divisible designs are introduced to settle this problem completely.     

Linked systems of symmetric group divisible designs

We introduce the concept of linked systems of symmetric group divisible designs. The connection with association schemes is established, and as a consequence we obtain an upper bound on the number of symmetric group divisible designs which are linked. Several examples of linked systems of symmetric group divisible designs are provided.     

Some new group divisible designs with block size 4 and two or three group sizes

Group divisible designs (GDDs) with block size 4 and at most 30 points are known for all feasible group types except three, namely $2^3{5}^4,3^5{6}^2$, and $2^2{5}^5$. In this paper we provide solutions for the first two of these three 4‐GDDs without assuming any automorphisms. We also construct several other 4‐GDDs. These include classes of 4‐GDDs of types ${\left(3m\right)}^4{\left(6m\right)}^q{\left(3n\right)}^1$ for $0\le n\le \left(q+1\right)m$ where $q\in \left\{2,3\right\}$ and solutions for 4‐GDDs of types $3^t{6}^s$ for a wide range of values of $s\ge 2,t\ge 4$ satisfying $t\equiv 0$ or $1\left(\mathrm{mod}4\right)$, including all cases with $4\le t\le s?1$. Most of the remaining unknown 4‐GDDs of type $3^t{6}^s$ have $\left(s?1\right)<t<2\left(s?1\right)$.     

A class of group divisible 3‐designs and their applications

In this article, we first show that a group divisible 3‐design with block sizes from {4, 6}, index unity and group‐type 2^m exists for every integer m≥ 4 with the exception of m = 5. Such group divisible 3‐designs play an important role in our subsequent complete solution to the existence problem for directed H‐designs DH_λ(m, r, 4, 3)s. We also consider a way to construct optimal codes capable of correcting one deletion or insertion using the directed H‐designs. In this way, the optimal single‐deletion/insertion‐correcting codes of length 4 can be constructed for all even alphabet sizes. © 2008 Wiley Periodicals, Inc. J Combin Designs 17: 136–146, 2009     

Incomplete group divisible designs with block size four

The object of this paper is the construction of incomplete group divisible designs (IGDDs) with block size four, group‐type (g, h)^u and index unity. It is shown that the necessary conditions for the existence of such an IGDD are also sufficient with three exceptions and six possible exceptions. © 2003 Wiley Periodicals, Inc. J Combin Designs 11: 442–455, 2003; Published online in Wiley InterScience ( www.interscience.wiley.com ). DOI 10.1002/jcd.10055