Composite construction of group divisible designs

Two new methods of constructing group divisible designs are given. In particular, a new resolvable solution for the SR 39 is presented.     

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     

Symmetry and classification of certain regular group divisible designs

In this paper it is shown that a regular group divisible (GD) design, with parametersv, b, r, k, ₁, ₂ satisfyingrk – ₂ v + 1 and ₂ = ₁ + 1, must be symmetric (i.e.,v + b). Furthermore, the parameters of such symmetric regular GD designs can be expressed in terms of only two integral parameters.Supported in part by Grant 59540043 (C), Japan.     

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     

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).     

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     

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.     

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.     

The Construction of Association Schemes by Parameter Amendment

Association scheme is a structure on a finite set that has some special relations among elements in the set. These relations are usually hidden in other relations, so how to derive them out is a problem. The paper gives a constructing method of mending parameters of the association schemes, and a new family of association schemes is obtained： quasiFmetric association scheme.     

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 .     

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.     

On the connectedness of partially balanced block designs

The necessary and sufficient conditions for m-associate partially balanced block (PBB) designs to be connected are given. This generalizes the criterion for m-associate partially balanced incomplete block (PBIB) designs, which has originally been established by Ogawa, Ikeda and Kageyama (1984, Proceedings of the Seminar on Combinatorics and Applications, 248–255, Statistical Publishing Society, Calcutta).This work was partially supported by the Polish Academy of Sciences Grant No. MR I.1-2/2.     

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 $\left(s?1\right)<t<2\left(s?1\right)$.     

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.     

The spectrum of additive BIB designs

In this article, the existence of additive BIB designs is discussed with direct and recursive constructions, together with investigation of a property of resolvability. Such designs can be used to construct infinite families of BIB designs. In particular, we obtain a series of B(sⁿ, ts^m, λt (ts^m ? 1) (s^n‐m ? 1)/[2(s^m ? 1)]) for any positive integer λ, such that sⁿ (sⁿ ? 1) λ ≡ 0 (mod s^m (s^m ? 1) and for any positive integer t with 2 ≤ t ≤ s^n‐m, where s is an odd prime power. Connections between additive BIB designs and other combinatorial objects such as multiply nested designs and perpendicular arrays are discussed. A construction of resolvable BIB designs with v = 4k is also proposed. © 2007 Wiley Periodicals, Inc. J Combin Designs 15: 235–254, 2007     

Group divisible designs with block size 4 where the group sizes are congruent to 2 mod 3

In this paper, we construct a number of 4-GDDs where the group sizes are all congruent to 2 (mod 3). We also show that 4-GDDs of type $2^t{8}^s$ exist for all but a finite number of feasible values of s and t. The largest unknown case has type $2^4{8}^{18}$ and has 152 points. A number of 4-GDDs with at most 50 points are also constructed. These include one of type $4^8{1}^1{10}^1$, the last feasible type of the form $4^s{1}^t{n}^1$ with at most 50 points for which no 4-GDD was known.     

Connectedness of PBIB designs having asymmetrical association schemes

Summary A necessary and sufficient condition for the connectedness ofm-associate partially balanced incomplete block (PBIB) designs having an asymmetrical association scheme is given, only in terms of design parameters, without inner structure parameters of designs. Supported in part by Grant 321-6066-58530013 (Japan).