Characterization of SDP Designs That Yield Certain Spin Models

We characterize the SDP designs that give rise to four-weight spin models with two values. We prove that the only such designs are the symplectic SDP designs. The proof involves analysis of the cardinalities of intersections of four blocks.AMS classification:05B20, 05E30     

平衡区组正交表与正交表的比较及应用

对平衡区组正交表和正交表进行的数据分析作了比较,表现出平衡区组正交表的优良性.给出了平衡区组正交表的应用实例,且对试验结果进行了分析.同时得出,分别用平衡区组正交表GL6(3221)和正交表L9(34)处理同一个问题时,所得试验结果一致.     

正交平衡区组设计统计模型中的方差分析

正交平衡区组设计(或广义正交表)的数据分析类似于正交拉丁方(或正交表)的数据分析.利用类似于正交表数据分析中的投影矩阵的正交分解技术,研究正交平衡区组设计的统计分析模型,给出了方差分析中的二次型以及各因子的二次型的分布性质,从而给出正交平衡区组设计统计模型中的方差分析方法.     

Alias balanced and alias partially balanced fractional 2 m factorial designs of resolution 2l+1

As a generalization of alias balanced designs due to Hedayat, Raktoe and Federer [5], we introduce the concept of alias partially balanced designs for fractional 2 ^m factorial designs of resolution 2l+1. All orthogonal arrays of strength 2l yield alias balanced designs. Some balanced arrays of strength 2l yield alias balanced and alias partially balanced designs. In particular, simple arrays which are a special case of balanced arrays yield alias partially balanced designs. At most 2 ^m −1 alias balanced (or alias partially balanced) designs are generated from an alias balanced (or alias partially balanced) design by level permutations. This implies that alias balanced or alias partially balanced designs need not be orthogonal arrays or balanced arrays of strength 2l.     

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.     

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.     

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.     

Robustness of row-column designs

Robustness of designs eliminating heterogeneity in two directions to outliers is studied. The important class of variance balanced row-column designs which satisfy the property of adjusted orthogonality are shown to be robust. Some general three-way balanced designs with balanced column vs row classification are also shown to be robust.     

正交平衡区组设计统计分析模型的参数估计

正交平衡区组设计(或者广义正交表)的数据分析类似于正交拉丁方(或者正交表)的数据分析,但试验次数大幅减少.引入了相遇平衡区组设计矩阵象的概念,定义了一种基于正交相遇平衡区组设计(或者广义正交表)的统计分析模型,根据这个模型,推导得到了参数的最小二乘估计.     

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.     

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     

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对试验次数较小的情形, 我们可以运用穷举法来寻找其可能存在的正交平衡区组设计. 这样做一方面可以尽可能多地找到正交平衡区组设计, 另一方面也可以为今后我们构造试验次数较大的正交平衡区组设计奠定基础. 本文最后给出了试验次数n=9以内的部分正交平衡区组设计.     

On construction of balancedn-ary block designs

Summary A method of construction of balanced ternary designs using affine α-resolvable balanced incomplete block designs is presented.     

Optimal Designs for Balanced Linear Mixed-Effects Regression Models

The paper investigates the problem of optimal balanced designs in general linear regression models with mixed effects. The interest lies in estimating fixed effects, random effects and prediction of the future observation of an individual, respectively. By using the de la Garaz phenomenon and Loewner order domination, the dimension of determining the optimal designs are reduced. The optimal designs are derived by using analytical or numerical methods, and their optimalities are verified through the general equivalence theorems.     

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)。     

Balanced arrays of strength two from block designs

Some constructions of balanced arrays of strength two are provided by use of rectangular designs, group divisible designs, and nested balanced incomplete block designs. Some series of such arrays are also presented as well as orthogonal arrays, with illustrations. © 2002 Wiley Periodicals, Inc. J Combin Designs 10: 303–312, 2002; Published online in Wiley InterScience ( www.interscience.wiley.com ). DOI 10.1002/jcd.10016     

Block designs admitting regular automorphisms

This paper treats a class of combinatorial designs which are essentially partially balanced incomplete block designs with two associate classes and with the additional feature that there are constants s and t so that for any treatment-block pair (x, B), the number of first associates of x in the block B is s or t depending on whether x ? B or x ? B. We obtain necessary and sufficient conditions that a P.B.I.B.D. should have this feature and obtain generalizations of earlier results on triangular and L₂ (n) P.B.I.B.D.'s. Finally, we consider two graphs naturally associated with quasi symmetric P.B.I.B.D.'s having this special feature determine their spectra, and investigate the conditions under which they are strongly regular.     

Exponentially Many Hadamard Designs

If there is a Hadamard design of order n, then there are at least 2^{8n−16−9log n} non-isomorphic Hadamard designs of order 2n. Mathematics Subject Classificaion 2000: 05B05     

正交平衡区组设计统计分析模型参数估计的矩阵表达

正交平衡区组设计(或者广义正交表)是一种类似于正交拉丁方(或者正交表)的新设计,但试验次数大幅减少.定义了一种基于正交相遇平衡区组设计(或者广义正交表)的统计分析模型,根据这个模型,给出了参数的最小二乘估计的矩阵形式.     

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.