Decomposable symmetric designs

The first infinite families of symmetric designs were obtained from finite projective geometries, Hadamard matrices, and difference sets. In this paper we describe two general methods of constructing symmetric designs that give rise to the parameters of all other known infinite families of symmetric designs. The method of global decomposition produces an incidence matrix of a symmetric design as a block matrix with each block being a zero matrix or an incidence matrix of a smaller symmetric design. The method of local decomposition represents incidence matrices of a residual and a derived design of a symmetric design as block matrices with each block being a zero matrix or an incidence matrix of a smaller residual or derived design, respectively.     

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.     

Matrix Image Method for Ranking Nonregular Fractional Factorial Designs

Nonregular fractional factorial designs such as Plackett-Burman designs and other orthogonal arrays are widely used in various screening experiments for their run size economy and flexibility. In this paper, we study matrix image theory and present a new method for distinguishing and assessing nonregular designs with complex alias structure, which works for all symmetrical and asymmetrical, regular and nonregular orthogonal arrays. Based on the matrix image theory, our proposed method captures orthogonality and projection properties. Empirical studies show that the proposed method has a more precise differentiation capacity when comparing with some other criteria.     

Completely Regular Designs of Strength One

We study a class of highly regular t-designs. These are the subsets of vertices of the Johnson graph which are completely regular in the sense of Delsarte [2]. In [9], Meyerowitz classified the completely regular designs having strength zero. In this paper, we determine the completely regular designs having strength one and minimum distance at least two. The approach taken here utilizes the incidence matrix of (t+1)-sets versus k-sets and is related to the representation theory of distance-regular graphs [1, 5].     

Combinatorial structures in loops I. Elements of the decomposition theory

Difference sets have been extensively studied in groups, principally in Abelian groups. Here we extend the notion of a difference set to loops. This entails considering the class of 〈υ, k〉 systems and the special subclasses of 〈υ, k, λ〉 principal block partial designs (PBPDs) and 〈υ, k, λ〉 designs. By means of a certain permutation matrix decomposition of the incidence matrices of a system and its complement, we can isomorphically identify an abstract 〈υ, k〉 system with a corresponding system in a loop. Special properties of this decomposition correspond to special algebraic properties of the loop. Here we investigate the situation when some or all of the elements of the loop are right inversive. We identify certain classes of 〈υ, k, λ〉 designs, including skew-Hadamard designs and finite projective planes, with designs and difference sets in right inverse property loops and prove a universal existence theorem for 〈υ, k, λ〉 PBPDs and corresponding difference sets in such loops.     

广义正交表正交性的等价条件

广义正交表是一种类似于正交表的新设计.正交平衡性是广义正交表必须满足的基本要求之一,它是正交表正交性的推广,它能够使得试验因子在方差分析中保持柯赫伦定理成立,因而可以像正交表一样进行试验设计和方差分析,从而不但保证其数据分析模型符合"不自生"逻辑,而且也可以保证试验因子的各种关系比较的数据分析结论具有客观一致性和可重复再现性,但试验次数大幅减少.利用矩阵象技术,提出并证明了广义正交表的组合正交性不但等价于其矩阵象的正交性,而且也等价于其广义关联矩阵的正交性.借助于SAS软件可以方便快速的验证某些区组设计相应的行列设计是否为广义正交表.     

Bounds for the number of common symbols in balanced and certain partially balanced designs

We secure inequalities on submatrices of N′N, where N is the incidence matrix of certain designs. These inequalities are then compared with other results previously known.     

A relation between BIB designs and chemical balance weighing designs

The paper gives certain new construction method for optimum chemical balance weighing designs. It utilizes a relation between the incidence matrices of a set of BIB designs and the design matrix of a chemical balance weighing design.     

On the non-existence of a class of symmetric group divisible partial designs

Many non-existence theorems are known for symmetric group divisible partial designs. In the case that these partial designs are auto-dual with ₁=0, an ideal incidence structure can be defined whose elements are the equivalence-classes of non-collinear points and parallel blocks. Except for some trivial cases this incidence structure turns out to be a symmetric design, and by studying its existence we can prove much more powerful non-existence theorems.     

Self-Orthogonal 3-(56,12,65) Designs and Extremal Doubly-Even Self-Dual Codes of Length 56

In this paper, we show that the code generated by the rows of a block-point incidence matrix of a self-orthogonal 3-(56,12,65) design is a doubly-even self-dual code of length 56. As a consequence, it is shown that an extremal doubly-even self-dual code of length 56 is generated by the codewords of minimum weight. We also demonstrate that there are more than one thousand inequivalent extremal doubly-even self-dual [56,28,12] codes. This result shows that there are more than one thousand non-isomorphic self-orthogonal 3-(56,12,65) designs. AMS Classification: 94B05, 05B05     

Divisible design graphs

A divisible design graph is a graph whose adjacency matrix is the incidence matrix of a divisible design. Divisible design graphs are a natural generalization of (v,k,λ)-graphs, and like (v,k,λ)-graphs they make a link between combinatorial design theory and algebraic graph theory. The study of divisible design graphs benefits from, and contributes to, both parts. Using information of the eigenvalues of the adjacency matrix, we obtain necessary conditions for existence. Old results of Bose and Connor on symmetric divisible designs give other conditions and information on the structure. Many constructions are given using various combinatorial structures, such as (v,k,λ)-graphs, distance-regular graphs, symmetric divisible designs, Hadamard matrices, and symmetric balanced generalized weighing matrices. Several divisible design graphs are characterized in terms of the parameters.     

Replication results for certain combinatorial configurations

Using matrix theoretic methods, bounds are obtained on the number of replications in certain combinatorial designs, some of which generalize previous results for multiplicative designs and λ-designs.     

On 2-(45, 12, 3) designs

Under the assumption that the incidence matrix of a 2-(45, 12, 3) design has a certain block structure, we determine completely the number of nonisomorphic designs involved. We discover 1136 such designs with trivial automorphism group. In addition we analyze all 2-(45, 12, 3) designs having an automorphism of order 5 or 11. Altogether, the total number of nonisomorphic 2-(45, 12, 3) designs found in 3752. Many of these designs are self-dual and each of these self-dual designs possess a polarity. Some have polarities with no absolute points, giving rise to strongly regular (45, 12, 3, 3) graphs. In total we discovered 58 pairwise nonisomorphic strongly regular graphs, one of which has a trivial automorphism group. Further, we analyzed completely all the designs for subdesigns with parameters 2-(12, 4, 3), 2-(9, 3, 3), and 2-(5, 4, 3). In the first case, the number of 2-(12, 4, 3) subdesigns that a design possessed, if non-zero, turned out to be a multiple of 3, whereas 2-(9, 3, 3) subdesigns were so abundant it was more unusual to find a design without them. Finally, in the case of 2-(5, 4, 3) subdesigns there is a design, unique amongst the ones discovered, that has precisely 9 such subdesigns and these form a partition of the point set of the design. This design has a transitive group of automorphisms of order 360. © 1996 John Wiley & Sons, Inc.     

Point-weight designs with design conditions on t points

This paper examines some of the properties of point-weight incidence structures, i.e. incidence structures for which every point is assigned a positive integer weight. In particular it examines point-weight designs with a design condition that stipulates that any two “identical” sets of t points must lie on the same number of blocks. We introduce a new class of designs with this property: row-sum designs, and examine the basic properties of row-sum point-weight designs and their similarities to classical (non-point-weight) designs and the point-weight designs of Horne [On point-weighted designs, Ph.D. Thesis, Royal Holloway, University of London, 1996].     

Some notes on integral equivalence of combinatorial matrices

Recent work on integral equivalence of Hadamard matrices and block designs is generalized in two directions. We first determine the two greatest invariants under integral equivalence of the incidence matrix of a symmetric balanced incomplete block design. This enables us to write down all the invariants in the case wherek −λ is square-fre. Some other results on the sequence of invariants are presented. Secondly we consider the existence of inequivalent Hadamard matrices under integral equivalence. We show that if there is a skew-Hadamard matrix of order 8m then there are two inequivalent Hadamard matrices of order 16m, that and there are precisely eleven inequivalent Hadamard matrices of order 32.     

Modular Hadamard martrices and related designs

A square matrix with entries ± 1 is called a modular Hadamard matrix if the inner product of each two distinct row vectors is a multiple of some fixed (positive) integer. This paper initiates the study of modular Hadamard matrices and the combinatorial designs associated with them. The related combinatorial designs are the main concern of this paper; some results dealing with the existence and construction of modular Hadamard matrices will be included in a later paper.     

Canonical forms of some special matrices useful in statistics

In experimental situations wheren two or three level factors are involved andn observations are taken, then theD-optimal first order saturated design is ann ×n matrix with elements ±1 or 0, ±1 with the maximum determinant. Canonical forms are useful for the specification of the non-isomorphicD-optimal designs. In this paper, we study canonical forms such as the Smith normal form, the first, second and the Jordan canonical form ofD-optimal designs. Numerical algorithms for the computation of these forms are described and some numerical examples are also given.     

On coloured constant composition designs

This paper deals with a particular class of coloured designs: the incidence vectors of all blocks of these designs have the same composition, and the same is true for the incidence vectors of all points. For this reason, we call these designs constant composition designs, or CC-designs for short. We will derive necessary and sufficient conditions on the existence and conclude the presentation with a collection of examples.     

Efficient experimental designs when most treatments are unreplicated

In early generation variety trials, large numbers of new breeders’ lines (varieties) may be compared, with each having little seed available. A so-called unreplicated trial has each new variety on just one plot at a site, but includes several replicated control varieties, making up around 10% and 20% of the trial. The aim of the trial is to choose some (usually around one third) good performing new varieties to go on for further testing, rather than precise estimation of their mean yields.Now that spatial analyses of data from field experiments are becoming more common, there is interest in an efficient layout of an experiment given a proposed spatial analysis and an efficiency criterion. Common optimal design criteria values depend on the usual C-matrix, which is very large, and hence it is time consuming to calculate its inverse. Since most varieties are unreplicated, the variety incidence matrix has a simple form, and some matrix manipulations can dramatically reduce the computation needed. However, there are many designs to compare, and numerical optimisation lacks insight into good design features. Some possible design criteria are discussed, and approximations to their values considered. These allow the features of efficient layouts under spatial dependence to be given and compared.     

A generalization of combinatorial designs and related codes

In this paper we study a certain generalization of combinatorial designs related to almost difference sets, namely the t-adesign, which was coined by Ding (Codes from difference sets, 2015). It is clear that 2-adesigns are partially balanced incomplete block designs which naturally arise in many combinatorial and statistical problems. We discuss some of their basic properties and give several constructions of 2-adesigns (some of which correspond to new almost difference sets and some to new almost difference families), as well as two constructions of 3-adesigns. We discuss basic properties of the incidence matrices and make an initial investigation into the codes which they generate. We find that many of the codes have good parameters in the sense they are optimal or have relatively high minimum distance.