New designs from circular nearrings

Nearrings are generalized rings in which addition is not in general abelian and only one distributive law holds. Some interesting combinatorial structures, as tactical configurations and balanced incomplete block designs (BIBDs) naturally arise when considering the class of planar and circular nearrings. We define the concept of disk and prove that in the case of field-generated planar circular nearrings it yields a BIBD. Such designs can be used in the construction of some classes of codes for which we are able to calculate the parameters.     

PBIBDs from weakly divisible nearrings and related codes

In [5] the authors are able to give a method for the construction of a family of partially balanced incomplete block designs from a special class of wd-nearrings (wd-designs). In this paper the wd-design incidence matrix and the connected row and column codes are studied. The parameters of two special classes of wd-designs and those of the related row and column codes are calculated.     

Virtually balanced incomplete block designs for v = 22, k = 8, and λ = 4

When basic necessary conditions for the existence of a balanced incomplete block design are satisfied, the design may still not exist or it may not be known whether it exists. In either case, other designs may be considered for the same parameters. In this article we introduce a class of alternative designs, which we will call virtually balanced incomplete block designs. From a statistical point of view these designs provide efficient alternatives to balanced incomplete block designs, and from a combinatorial point of view they offer challenging new questions. © 1995 John Wiley & Sons, Inc.     

Double Arrays,Triple Arrays and Balanced Grids

Triple arrays are a class of designs introduced by Agrawal in 1966 for two-way elimination of heterogeneity in experiments. In this paper we investigate their existence and their connection to other classes of designs, including balanced incomplete block designs and balanced grids.AMS Classification:05B05, 05B30     

On the incomplete block designs derivable from the association schemes

Blackwelder (1969) has given two methods of constructing balanced incomplete block (BIB) designs from the association matrices of association schemes with two and three associate classes. In this note these two methods are incorporated in a general method, and the existence of a series of BIB designs is shown by the generalized method. In addition, a remark about partially balanced incomplete block (PBIB) designs with respect to the method is made.     

Some existence and construction results of polygonal designs

This paper revisits the existence and construction problems for polygonal designs (a special class of partially balanced incomplete block designs associated with regular polygons). We present new polygonal designs with various parameter sets by explicit construction. In doing so we employ several construction methods — some conventional and some new. We also establish a link between a class of polygonal designs of block size 3 and the cyclically generated ‘λ-fold triple systems’. Finally, we show that the existence question for a certain class of polygonal designs is equivalent to the existence question for ‘perfect grouping systems’ which we introduce.     

A class of pbib-designs obtained from projective geometries

In this paper, a new class of partially balanced incomplete block designs is constructed over an association scheme obtained from a finite projective geometry. Further, a general method for deriving a balanced incomplete block design from another one is given.     

The 2-adjugate mod 2 class of designs

In this paper we define a 2-adjugate mod 2 class of designs obtained by application of process of 2-adjugation to the incidence matrix of given design and subsequently reducing to modulo 2. General properties of such designs are discussed and its application to the class of unreduced Balanced Incomplete Block Designs is investigated in detail. It is found that in this case we obtain a class of generalized partially balanced incomplete block designs with 2 associate classes and with triangular association scheme, but with unequal block sizes.     

Splitting balanced incomplete block designs with block size 3 × 2

Splitting balanced incomplete block designs were first formulated by Ogata, Kurosawa, Stinson, and Saido recently in the investigation of authentication codes. This article investigates the existence of splitting balanced incomplete block designs, i.e., (v, 3k, λ)‐splitting BIBDs; we give the spectrum of (v, 3 × 2, λ)‐splitting BIBDs. As an application, we obtain an infinite class of 2‐splitting A‐codes. © 2004 Wiley Periodicals, Inc.     

某些可分解平衡不完全区组设计

本文证明存在某些可分解平衡不完全区组设计RB(k,λ;v)。例如,存在RB(6,5;42)及RB(5,2;55)。     

A class of three-replicate three-associate p.b.i.b. designs

Summary Partially balanced incomplete block (p.b.i.b.) designs of two and more associate classes have been constructed by Saha and Das [9] through the use of confounded designs for 2 ⁿ factorals. By dualising one series of two-associated p.b.i.b. designs given by them, a class of three-replicate three-associate p.b.i.b. designs have been obtained. This can be taken to be a new class of designs in the sense that they are not included by Nair [7] in his table of three-replicate three-associate p.b.i.b. designs.     

Optimal balanced 27 fractional factorial designs of resolutionV,withN≦42

Summary In this paper, we present a class of fractional factorial designs of the 2⁷ series, which are of resolutionV. Such designs allow the estimation of the general mean, the main effects and the two factors interactions (29 parameters in all for the 2⁷ factorial) assuming that the higher order effects are negligible. For every value ofN (the number of runs) such that 29≦N≦42, we give a resolutionV design that is optimal (with respect to the trace criterion) within the subclass of balanced designs. Also, for convenience of analysis, we present for each design, the covariance matrix of the estimates of the various parameters. As a by product, we establish many interesting combinatorial theorems concerning balanced arrays of strength four (which are generalizations of orthogonal arrays of strength four, and also of balanced incomplete block designs with block sizes not necessarily equal).     

On the optimality of a class of designs with three concurrences

In the present paper we consider a class of unequally replicated designs having concurrence range 2 and spectrum of the form μ₁(μ₂)^v−3μ₃. Now, Jacroux’s [Some sufficient conditions for the type I optimality of block designs, J. Statist. Plann. Inference 11 (1985) 385-396] Proposition 2.4 says that a design with spectrum of the above form, if satisfies some further conditions, is type 1 optimal. Unfortunately, this proposition does not apply to our designs since they have a poor status regarding E-optimality. Yet we are able to prove the A-optimality (in the general class) of these designs using majorisation technique. A method of construction of an infinite series of our A-optimal designs has also been given.The first and only known infinite series of examples of designs satisfying Jacroux’s conditions appears to be the first one in Section 4.1 of Morgan and Srivastav [On the Type-1 optimality of nearly balanced incomplete block designs with small concurrence range, Statist. Sinica 10 (2000) 1091-1116] - hitherto referred to as [MS]. In this paper, we use majorisation technique to prove stronger optimality properties of the above mentioned designs of [MS] as well as to present simpler proof of another optimality result in [MS].     

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.     

两类组合设计及其在认证码中的应用

W．Ogata等定义了两种新的组合设计：外差族（EDF）与外平衡不完全区组设计（E-BIBD）．本文首先用有限域中的分圆类给出EDF的一个构造；接着用EBIBD构造出具有完善保密性的最优分裂A-码，然后证明了由满足一定条件的两个EBIBD通过上述方法构造出的两个认证码是同构的.     

Construction of nested balanced block designs, rectangular designs andq-ary codes

Construction of nested balanced incomplete block (BIB) designs, nested balanced ternary designs and rectangular designs from given nested BIB designs and resolvable BIB designs are described. New constructions ofq-ary codes from nested BIB designs and balanced bipartite weighing designs are also given.     

Partial geometric designs and two-class partially balanced designs

It is shown that a partial geometric design with parameters (r, k, t, c) satisfying certain conditions is equivalent to a two-class partially balanced incomplete block design. This generalizes a result concerning partial geometric designs and balanced incomplete block designs.     

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.     

EMBEDDINGS OF SIMPLE TWO-FOLD BALANCED INCOMPLETE BLOCK DESIGNS WITH BLOCK SIZE FOUR

The necessary and sufficient conditions for the existence of simple incomplete block design (v, w; 4,2)-IPBDs are determined. As a consequence, the necessary and sufficient conditions for the embeddings of simple two-fold balanced incomplete block designs with block size 4 are also determined.     

Balanced Incomplete Block Designs with Block Size 9 and λ=2,4,8

The necessary conditions for the existence of a balanced incomplete block design on v points, with index λ and block size k, are that: $$\begin{gathered} {\text{ }}\lambda (v - 1) \equiv 0{\text{ mod (}}k - 1{\text{)}} \hfill \\ \lambda v(v - 1) \equiv 0{\text{ mod (}}k - 1{\text{)}} \hfill \\ \end{gathered} $$ In this paper we study k=9 with λ=2,4 or 8. For λ=8, we show these conditions on v are sufficient, and for λ=2, 4 respectively there are 8 and 3 possible exceptions the largest of which are v=1845 and 783. We also give some examples of group divisible designs derived from balanced ternary designs.