Algebraic duality theorems with combinatorial applications

We state and prove two rather direct linear algebra results and show how they are the basis for many combinatorial devices for determining the dual structure of combinatorial designs. Applications to tactical decompositions and partial geometric designs are also included.     

Constructions of External Difference Families and Disjoint Difference Families

External difference families (EDFs) are a type of new combinatorial designs originated from cryptography. In this paper, some earlier ideas of recursive and cyclotomic constructions of combinatorial designs are extended, and a number of classes of EDFs and disjoint difference families are presented. A link between a subclass of EDFs and a special type of (almost) difference sets is set up.     

Construction of optimal supersaturated designs by the packing method

A supersaturated design is essentially a factorial design with the equal occurrence of levels property and no fully aliased factors in which the number of main effects is greater than the number of runs. It has received much recent interest because of its potential in factor screening experiments. A packing design is an important object in combinatorial design theory. In this paper, a strong link between the two apparently unrelated kinds of designs is shown. Several criteria for comparing supersaturated designs are proposed, their properties and connections with other existing criteria are discussed. A combinatorial approach, called the packing method, for constructing optimal supersaturated designs is presented, and properties of the resulting designs are also investigated. Comparisons between the new designs and other existing designs are given, which show that our construction method and the newly constructed designs have good properties.     

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.     

Incidence matrices and inequalities for combinatorial designs

In this paper we use incidence matrices of block designs and row–column designs to obtain combinatorial inequalities. We introduce the concept of nearly orthogonal Latin squares by modifying the usual definition of orthogonal Latin squares. This concept opens up interesting combinatorial problems and is expected to be useful in planning experiments by statisticians. © 2002 John Wiley & Sons, Inc. J Combin Designs 10: 17–26, 2002     

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 squashed design

Extremal problems and the existence of designs is investigated in a new type of combinatorial structures, called squashed geometries.     

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.     

High-rate LDPC codes from partially balanced incomplete block designs

Journal of Algebraic Combinatorics - This paper presents a combinatorial construction of low-density parity-check (LDPC) codes from partially balanced incomplete block designs. Since...     

混水平均匀设计的构造

我们用离散偏差来度量部分因子设计的均匀性,本文的目的在于寻找一些构造混水平均匀设计的方法,这些方法比文献中已有的方法更简单且计算成本更低.我们得到了离散偏差的一个下界,如果一个U 型设计的离散偏差值达到这个下界,那么该设计是—个均匀设计.我们建立了均匀设计与组合设计理论中一致可分解设计之间的联系.通过一致可分解设计,我们提出了一些构造均匀设计的新方法,同时也给出了许多均匀设计存在的无穷类.     

Constructions of uniform designs by using resolvable packings and coverings

Uniform designs have been widely used in computer experiments, as well as in industrial experiments when the underlying model is unknown. Based on the discrete discrepancy, the link between uniform designs, and resolvable packings and coverings in combinatorial theory is developed. Through resolvable packings and coverings without identical parallel classes, many infinite classes of new uniform designs are then produced.     

Twofold triple systems and graph imbeddings

A natural bijection between (v, 3, 2)-BIBD's and certain topological and combinatorial structures related to graph imbeddings is used to obtain new classes of designs of this type. The topological viewpoint often makes it quite simple to differentiate between nonisomorphic designs.     

The first families of highly symmetric Kirkman Triple Systems whose orders fill a congruence class

Designs, Codes and Cryptography - Kirkman triple systems (KTSs) are among the most popular combinatorial designs and their existence has been settled a long time ago. Yet, in comparison with...     

Generalized covering designs and clique coverings

Inspired by the “generalized t‐designs” defined by Cameron [P. J. Cameron, Discrete Math 309 (2009), 4835–4842], we define a new class of combinatorial designs which simultaneously provide a generalization of both covering designs and covering arrays. We then obtain a number of bounds on the minimum sizes of these designs, and describe some methods of constructing them, which in some cases we prove are optimal. Many of our results are obtained from an interpretation of these designs in terms of clique coverings of graphs. © 2011 Wiley Periodicals, Inc. J Combin Designs 19:378‐406, 2011     

最优带仲裁认证码

本文研究带仲裁认证码（亦称A^2-码）。一个A^2-码，当它的欺骗攻击成功概率达到信息论下界且密钥长度也达到最小时称为是最优的。本文给出最优A^2-码的组合结构的特征，从而将构造最优A^2-码的问题归结为构造相应的一类组合设计的问题。     

A note on the construction of certain BIB designs

In this note a method of construction of certain combinatorial designs is defined. This gives the solution of (121, 132, 60, 55, 27) which is marked as unknown by Kageyama [1].     

Minimum secondary aberration fractional factorial split-plot designs in terms of consulting designs

It is very powerful for constructing nearly saturated factorial designs to characterize fractional factorial (FF) designs through their consulting designs when the consulting designs are small. Mukerjee and Fang employed the projective geometry theory to find the secondary wordlength pattern of a regular symmetrical fractional factorial split-plot (FFSP) design in terms of its complementary subset, but not in a unified form. In this paper, based on the connection between factorial design theory and coding theory, we obtain some general and unified combinatorial identities that relate the secondary wordlength pattern of a regular symmetrical or mixed-level FFSP design to that of its consulting design. According to these identities, we further establish some general and unified rules for identifying minimum secondary aberration, symmetrical or mixed-level, FFSP designs through their consulting designs.     

Syzygies for Metropolis base chains

It is shown that Markov chains for sampling from combinatorial sets in the form of experimental designs can be made more efficient by using syzygies on gradient vectors. Examples are presented.     

Existence of Perfect 3-Deletion-Correcting Codes

Bours [4] recently showed some constructions for perfect 2 and 3-deletion-correcting codes from combinatorial designs. He settled existence of perfect 2-deletion-correcting codes with words of length 4. However, the existence of perfect 3-deletion-correcting codes with words of length 5, or T^*(2, 5, v), remained unsettled for v 7, 8 (mod 10) and v = 13, 14, 15, 16. In this paper we provide new constructions for these codes from combinatorial designs, and show that a T^*(2, 5, v) exists for all v.