Uniformly resolvable pairwise balanced designs with blocksizes two and three

A uniformly resolvable pairwise balanced design is a pairwise balanced design whose blocks can be resolved into parallel classes in such a way that all blocks in a given parallel class have the same size. We are concerned here with designs in which each block has size two or three, and we prove that the obvious necessary conditions on the existence of such designs are also sufficient, with two exceptions, corresponding to the non-existence of Nearly Kirkman Triple Systems of orders 6 and 12.     

Modified frames with block size 3

A modified (k, λ)‐frame of type g^u is a modified (k, λ)‐GDD whose blocks can be partitioned into holey parallel classes, each of which is with respect to some group. Modified frames can be used to construct some other resolvable designs such as resolvable group divisible designs and semiframes. In this article, we shall investigate the existence of modified frames with block size 3. © 2007 Wiley Periodicals, Inc. J Combin Designs 16: 351–363, 2008     

Kirkman Packing Designs KPD ({3, 5*},v)

A Kirkman packing design KPD ({3, 5^*},v) is a resolvable packing with maximum possible number of parallel classes, each class containing one block of size 5 and all other blocks of size three. Such designs can be used to construct certain threshold schemes in cryptography. In this paper, direct and recursive constructions are discussed for such designs. The existence of a KPD ({3, 5^*},v) for is established with a few possible exceptions.     

Maximum uniformly resolvable designs with block sizes 2 and 4

A central question in design theory dating from Kirkman in 1850 has been the existence of resolvable block designs. In this paper we will concentrate on the case when the block size k=4. The necessary condition for a resolvable design to exist when k=4 is that v≡4mod12; this was proven sufficient in 1972 by Hanani, Ray-Chaudhuri and Wilson [H. Hanani, D.K. Ray-Chaudhuri, R.M. Wilson, On resolvable designs, Discrete Math. 3 (1972) 343-357]. A resolvable pairwise balanced design with each parallel class consisting of blocks which are all of the same size is called a uniformly resolvable design, a URD. The necessary condition for the existence of a URD with block sizes 2 and 4 is that v≡0mod4. Obviously in a URD with blocks of size 2 and 4 one wishes to have the maximum number of resolution classes of blocks of size 4; these designs are called maximum uniformly resolvable designs or MURDs. So the question of the existence of a MURD on v points has been solved for by the result of Hanani, Ray-Chaudhuri and Wilson cited above. In the case this problem has essentially been solved with a handful of exceptions (see [G. Ge, A.C.H. Ling, Asymptotic results on the existence of 4-RGDDs and uniform 5-GDDs, J. Combin. Des. 13 (2005) 222-237]). In this paper we consider the case when and prove that a exists for all u≥2 with the possible exception of u∈{2,7,9,10,11,13,14,17,19,22,31,34,38,43,46,47,82}.     

Two-level fractional factorial designs in blocks of size two for the orthogonal estimation of all main effects and two-factor interactions

In this paper we consider the problem of constructing two-level fractional factorial designs in blocks of size two that allow for the orthogonal estimation of all main effects and two-factor interactions (after adjusting for blocks). This problem has been considered in the literature, e.g., see Yang and Draper (2003), Wang (2004) and Kerr (2006). In this paper we give two systematic methods for the construction of such designs. The first construction method gives in many situations designs requiring fewer runs than those designs previously given whereas the second method gives a systematic method for constructing designs analogous to those illustrated in Yang and Draper (2003) by example.     

The spectrum of α‐resolvable designs with block size four

Abstact: An α‐resolvable BIBD is a BIBD with the property that the blocks can be partitioned into disjoint classes such that every class contains each point of the design exactly α times. In this paper, we show that the necessary conditions for the existence of α‐resolvable designs with block size four are sufficient, with the exception of (α, ν, λ) = (2, 10, 2). © 2000 John Wiley & Sons, Inc. J Combin Designs 9: 1–16, 2001     

A simplified version of strong connectivity in geometries

If there are less than b distinct blocks in a BIB design with b blocks then we say the design has repeated blocks. The set of distinct blocks of a design is called the support of the design. BIB designs with repeated blocks, besides being optimal, have special applications in the design of experiments and controlled samplings. Construction of BIB(ν, b, r, k, λ) designs with repeated blocks becomes complicated whenever the three parameters b, r, and λ are relatively prime. BIB(8, 56, 21, 3, 6) designs are examples of such designs with the smallest number of varieties. BIB(10, 30, 9, 3, 2) designs are such designs with the smallest number of blocks. We make an interesting observation about BIB(8, 56, 21, 3, 6) designs and give a table of such designs with 30 different support sizes. We prove, by construction, that a BIB(10, 30, 9, 3, 2) design exists if and only if the support size belongs to {21, 23, 24, 25, 26, 27, 28, 29, 30}. Other results are also given.     

Pairwise and variance balanced incomplete block designs

Summary The purpose of this paper is three-fold. The first purpose is to compile and to systematize published and dispersed results on two aspects of balancing in incomplete block designs, i.e., pairwise balance and variance balance. This was done in order to establish the status of these two concepts of balance in published literature and to put them in a form which is useful for further work in this area. Also, the results in this form are necessary for the development of the remainder of the paper. The second purpose of this paper is to present a method of constructing unequal replicate and/or unequal block size experiment designs for which the variance balance property is achieved. The method of construction involves the union of blocks from two or more block designs and the augmentation of some of the blocks with additional treatments; the method is denoted asunionizing block designs. A straight-forward extension of the method would produce a partially balanced block design with unequal replicate and/or unequal block designs. The enlargement of the concept and availability of variance blanced block designs to accommodate unequal replication and/or unequal block sizes is important to the researcher, the teacher, and the experimenter needing such designs. For example, an animal nutritionist or a psychologist is no longer required to have constant litter or family sizes for the blocks and may have unequal replication on the treatments for those treatments with insufficient material and still attain the goal of equal variances on all normalized treatment contracts. The third purpose of the paper is to apply the unionizing block designs method to construct a family of unequal replicate and unequal block size variance balanced designs. Some comments are given on the extension of the unionizing block designs method to construct other families of variance balanced or partially balanced block designs. This investigation was supported in part by PHS Research Grant No. R01-GM-05900 from General Medical Services.     

Group divisible designs with three unequal groups and larger first index

We show the necessary conditions are sufficient for the existence of group divisible designs (or PBIBDs) with block size k=3 with three groups of size (n,2,1) for any n≥2 and any two indices with λ₁>λ₂.     

Steiner triple systems with two disjoint subsystems

It is shown that there exists a triangle decomposition of the graph obtained from the complete graph of order v by removing the edges of two vertex disjoint complete subgraphs of orders u and w if and only if u,w, and v are odd, (mod 3), and . Such decompositions are equivalent to group divisible designs with block size 3, one group of size u, one group of size w, and v – u – w groups of size 1. This result settles the existence problem for Steiner triple systems having two disjoint specified subsystems, thereby generalizing the well‐known theorem of Doyen and Wilson on the existence of Steiner triple systems with a single specified subsystem. © 2005 Wiley Periodicals, Inc. J Combin Designs     

Group divisible designs with three groups and block size four

We present new constructions and results on GDDs with three groups and block size four and also obtain new GDDs with two groups of size nine. We say a GDD with three groups is even, odd, or mixed if the sizes of the non-empty intersections of any of its blocks with any of the three groups is always even, always odd, or always mixed. We give new necessary conditions for these families of GDDs and prove that they are sufficient for these three types and for all group sizes except for the minimal case of mixed designs for group size 5t(t>1). In particular, we prove that mixed GDDs allow a maximum difference between indices. We apply the constructions given to show that the necessary conditions are sufficient for all GDDs with three groups and group sizes two, three, and five, and also for group size four with two possible exceptions, a GDD(4,3,4;5,9) and a GDD(4,3,4;7,12).     

On the construction of handcuffed designs

A handcuffed design with parameters ν, k, λ consists of a set of ordered k-subsets of a v-set, called handcuffed blocks; in a block (a₁, a₂, a_k) each element is assumed to be “handcuffed” to its neighbors. A block, therefore, contains k ? 1 handcuffed pairs, the pairs being considered unordered. Each element of the v-set appears in exactly r blocks, and each pair of distinct elements of the v-set is handcuffed in exactly A blocks of the design.These designs have been studied recently by Hung and Mendelsohn [1], who construct a number of families of such designs by recursive methods. In this paper we show how difference methods can be applied to the construction of handcuffed designs. The methods are powerful, and a number of families of designs are constructed. A main new result is the determination of necessary and sufficient conditions for the existence of handcuffed designs for all parameter sets in which v is an odd prime power.     

A 2-(22, 8, 4) Design Cannot Have a 2-(10, 4, 4) Subdesign

The smallest BIBD, as for the number of points and blocks, whose existence is still undecided is 2-(22, 8, 4). Possible subconfigurations of such a design, namely 2-(10, 4, 4) designs, are here ruled out. The result is obtained by classifying all 2-(10, 4, 4) designs and trying to find 2-(22, 8, 4) designs by solving instances of the maximum clique problem.     

Balanced nested designs and balanced arrays

Balanced nested designs are closely related to other combinatorial structures such as balanced arrays and balanced n-ary designs. In particular, the existence of symmetric balanced nested designs is equivalent to the existence of some balanced arrays. In this paper, various constructions for symmetric balanced nested designs are provided. They are used to determine the spectrum of symmetric balanced nested balanced incomplete block designs with block size 3 and 4.     

Edge-coloured designs with block size four

Summary The existence of edge-coloured block designs with block size four is studied for all nonisomorphic colourings of the edges of aK ₄. There are 25 nonisomorphic edge-colouredK ₄'s; for each, we examine the existence of edge-coloured designs with the minimum possible index. Uniform cases lead to block designs, perpendicular arrays, nested Steiner triple systems, idempotent Schroeder quasigroups, and other combinatorial objects.     

More balanced ternary designs with block size three

A balanced ternary design (BTD) is a balanced design on V elements with constant block size, in which each element may occur 0, 1 or 2 times in each block. Thus blocks may be collections of elements rather than subsets of elements. A regular BTD is one in which each element occurs singly in ?₁ blocks, and is repeated in ?₂ blocks. The condition V ≡ 0(mod 3) was shown in an earlier paper to be both necessary and sufficient for the existence of a regular BTD on V elements with block size 3, index 2 and ?₂=1. We show in this paper that if ?₂ = 2, necessary and sufficient conditions for existence of a regular BTD with block size 3 and index 2 are V ≡0 or 2(mod 3), V ≥5, and if ?₂=3 the necesary and sufficient conditions are V ≡ 0 or 1(mod 3), V ≥ 7. Hence by allowing ?₂ to be as large as 3, we have a regular BTD on V elements with block size 3 and index 2 for all V ≠1,2,4.     

Class‐uniformly resolvable designs

We investigate Class‐Uniformly Resolvable Designs, which are resolvable designs in which each of the resolution classes has the same number of blocks of each size. We derive the fully general necessary conditions including a number of extremal bounds. We present two general constructions. We primarily consider the case of block sizes 2 and 3, where we find two infinite extremal families and finish two other infinite families by difference constructions. We present tables showing the current state of knowledge in the case of block size 2 and 3 for all orders up to 200. © 2001 John Wiley & Sons, Inc. J Combin Designs 8: 79–99, 2001     

Copula‐based robust optimal block designs

Blocking is often used to reduce known variability in designed experiments by collecting together homogeneous experimental units. A common modeling assumption for such experiments is that responses from units within a block are dependent. Accounting for such dependencies in both the design of the experiment and the modeling of the resulting data when the response is not normally distributed can be challenging, particularly in terms of the computation required to find an optimal design. The application of copulas and marginal modeling provides a computationally efficient approach for estimating population‐average treatment effects. Motivated by an experiment from materials testing, we develop and demonstrate designs with blocks of size two using copula models. Such designs are also important in applications ranging from microarray experiments to experiments on human eyes or limbs with naturally occurring blocks of size two. We present a methodology for design selection, make comparisons to existing approaches in the literature, and assess the robustness of the designs to modeling assumptions.     

Small Box-Behnken design

Box-Behnken design has been popularly used for the second-order response surface model. It is formed by combining two-level factorial designs with incomplete block designs in a special manner—the treatments in each block are replaced by an identical design. In this paper, we construct small Box-Behnken design. These designs can fit the second-order response surface model with reasonably high efficiencies but with only a much smaller run size. The newly constructed designs make use of balanced incomplete block design (BIBD) or partial BIBD, and replace treatments partly by 2_III³⁻¹ designs and partly by full factorial designs. It is shown that the orthogonality properties in the original Box and Behnken designs will be kept in the new designs. Furthermore, we classify the parameters into groups and introduce Group Moment Matrix (GMM) to estimate all the parameters in each group. This allows us to significantly reduce the amount of computational costs in the construction of the designs.     

自同构群基柱为~3D_4(q)的2-(v,k,1)设计

2-(v,k,1)设计的存在性问题是组合设计理论中重要的问题,当这类设计具有一个有意义自同构群时,讨论其存在性是尤其令人感兴趣的.30年前,一个6人团队基本上完成了旗传递的2-(v,k,1)设计分类.此后,人们开始致力于研究区传递但非旗传递的2-(v,k,1)设计的分类课题.本文证明了自同构群基柱为~3D_4(q)的区传递及点本原非旗传递的2-(v,k,1)设计是不存在的.