Asymmetric Orthogonal Arrays Research Articles

Construction of Asymmetric Orthogonal Arrays With High Strength

ABSTRACTOrthogonal arrays (OAs) are a key research focus in combinatorial design theory and experimental design theory. They play a very important role in an increasing number of interdisciplinary fields, such as computer science, cryptography, multipartite quantum states and quantum information. And the open problems in many fields are closely related to the construction of orthogonal arrays with high strength and large size. At present, some open problems have been solved by using orthogonal arrays. There is an urgent need for developing such orthogonal arrays. This article introduces a general method for obtaining asymmetric orthogonal arrays with high strength. The proposed method is straightforward and constructive. The constructed OAs can be utilized to generate more OAs. Some selective OAs are tabulated for practical applications.

Near MDS and near quantum MDS codes via orthogonal arrays

Near maximum distance separable (NMDS) codes are closely related to interesting objects in finite geometry and have nice applications in combinatorics and cryptography. But there are many unsolved problems about construction of NMDS codes. In this paper, by using symmetrical orthogonal arrays (OAs), we construct a lot of NMDS, m-MDS and almost extremal NMDS codes. Quantum error-correcting codes (QECCs) play a central role in quantum information processing and can protect quantum information from various quantum noises. We present a general method for constructing QECCs over mixed alphabets through asymmetrical OAs. Since quantum maximum distance separable (QMDS) codes over mixed alphabets with the dimension equal to one have not been found in all the literature so far, the definition of a near QMDS code over mixed alphabets is proposed. By using asymmetrical OAs, we obtain many such codes.

Construction of asymmetric orthogonal arrays of high strength via generator matrix

Asymmetric orthogonal arrays are very useful in theoretical studies and practical applications. However, it is difficult to construct such designs of high strength. In this article, we use the generator matrix method to obtain some asymmetric orthogonal arrays of strength three. Then we combine it with the general (further) replacement procedure to get more general asymmetric orthogonal arrays of high strength.

On the construction of some new asymmetric orthogonal arrays

Orthogonal arrays are of great significance in statistics and combinatorics. In this paper, we construct several new families of asymmetric orthogonal arrays of high strength by furthermore developing the existing methods.

Construction of mixed orthogonal arrays with high strength

A considerable portion of the work on mixed orthogonal arrays applies specifically to arrays of strength 2. Although strength t=2 is arguably the most important case for statistical applications, there is an urgent need for better methods for t≥3. However, the knowledge on the existence of arrays for t≥3 is rather limited. In this paper, new construction methods for symmetric and asymmetric orthogonal arrays (OAs) with high strength are proposed by using lower strength orthogonal partitions of spaces and OAs. A positive answer is provided to the open problem in Hedayat, Sloane and Stufken (Orthogonal Arrays: Theory and Applications (1999) Springer) on developing better methods and tools for the construction of mixed orthogonal arrays with strength t≥3. Not only are the methods straightforward, but also they are useful for constructing symmetric or asymmetric OAs of arbitrary strengths, numbers of levels and various sizes. The constructed OAs can be utilized to generate more OAs. The resulting OAs have a high degree of flexibility and many other desirable properties. Some selective OAs are tabulated for practical uses.

D-optimal augmented designs and the existence of tight orthogonal arrays with high strength

In this paper, in view of the Rao’s inequalities on the parameters of an orthogonal array, we define some extended matrices for asymmetric orthogonal arrays of strength t(t≥2). According to these matrices, necessary conditions for the existence of tight asymmetric orthogonal arrays of strengths 2 and 4 described by Mukerjee and Wu (1995) are generalized to the existence of tight asymmetric orthogonal arrays of strength t, and examples of the nonexistence of tight asymmetric orthogonal arrays with high strength are interpreted. Furthermore, some conditions on the additional p runs are determined, which guarantee that the orthogonal array plus p run design is D-optimal.

Column-orthogonal designs with multi-dimensional stratifications

The orthogonal Latin hypercube design and its relaxation, and column-orthogonal design, are two kinds of orthogonal designs for computer experiments. However, they usually do not achieve maximum stratifications in multi-dimensional margins. In this paper, we propose some methods to construct column-orthogonal designs with multi-dimensional stratifications by rotating symmetric and asymmetric orthogonal arrays. The newly constructed column-orthogonal designs ensure that the estimates of all linear effects are uncorrelated with each other and even uncorrelated with the estimates of all second-order effects (quadratic effects and bilinear effects) when the rotated orthogonal arrays have strength larger than two. Besides orthogonality, the resulting designs also preserve better space-filling property than those constructed by using the existing methods. In addition, we provide a method to construct a new class of orthogonal Latin hypercube designs with multi-dimensional stratifications by rotating regular factorial designs. Some newly constructed orthogonal Latin hypercube designs are tabulated for practical use.

Construction of asymmetric orthogonal arrays of high strength by juxtaposition

In this paper, we propose a juxtaposition method from one column to multi-column to construct asymmetric orthogonal arrays with high strength, which is a generalization of one column juxtaposition constructing asymmetric orthogonal arrays with strength 2. Five main theoretical results for the juxtaposition method are given. Many asymmetric orthogonal arrays of strength 3 and 4 are constructed by the juxtaposition method, including some new tight asymmetric orthogonal arrays of strength 3.

The Hamming distances of saturated asymmetrical orthogonal arrays with strength 2

Orthogonal arrays have many connections to other combinatorial designs and are applied in coding theory, the statistical design of experiments, cryptography, various types of software testing and quality control. In this paper, we present some general methods to find the Hamming distances for saturated asymmetrical orthogonal arrays (SAOAs) with strength 2. As applications of our methods, the Hamming distances of SAOA parents of size less than or equal to 100 are obtained. We also provide the Hamming distances of the SAOAs constructed from difference schemes or by the expansive replacement method. The feasibility of Hamming distances is discussed.

Construction of Symmetric and Asymmetric Orthogonal Arrays of Strength t from Orthogonal Partition

By using orthogonal partition, a general method to construct symmetric and asymmetric orthogonal arrays of strength t is proposed. And the orthogonal arrays constructed by the method are new.

Construction of Asymmetric Orthogonal Arrays of Strength <i>t</i> from Orthogonal Partition of Small Orthogonal Arrays

Construction of Asymmetric Orthogonal Arrays of Strength <i>t</i> from Orthogonal Partition of Small Orthogonal Arrays

Construction of Asymmetric Orthogonal Arrays of Strength Three via a Replacement Method

AbstractA replacement procedure to construct orthogonal arrays of strength three was proposed by Suen et al. [7]. This method was later extended by Suen and Dey [8]. In this paper, we further explore the replacement procedure to obtain some new families of orthogonal arrays of strength three.

ORTHOGONAL ARRAYS CONSTRUCTED BY GENERALIZED KRONECKER PRODUCT

In this paper, we propose a new general approach to construct asymmetrical orthogonal arrays, namely generalized Kronecker product. The operation is not usual Kronecker product in the theory of matrices, but it is interesting since the interaction of two columns of asymmetrical orthogonal arrays can be often written out by the generalized Kronecker product. As an application of the method, some new mixed-level orthogonal arrays of run sizes 72 and 96 are constructed.

On the construction of asymmetric orthogonal arrays

A general method of construction of asymmetric orthogonal arrays was proposed by Suen et al. (2001). Using this method and some modifications thereof, we construct some more families of asymmetric orthogonal arrays of strength greater than two.

Sliced space-filling designs with different levels of two-dimensional uniformity

We consider sliced computer experiments where priori knowledge suggests that factors may have different levels of importance, and so some factors need to be paid more attention than others. A new class of sliced space-filling designs are proposed to deal with this type of sliced computer experiments, in which the whole design and each slice may have different levels of two-dimensional uniformity for different factors, besides they all achieve maximum stratification in univariate margins. They are generated by elaborately randomizing a special type of asymmetric orthogonal arrays, called asymmetric balanced sliced orthogonal arrays, which can be partitioned into several slices such that each slice is balanced and becomes an asymmetric orthogonal array after some level-collapsing. Several methods are developed to construct such asymmetric balanced sliced orthogonal arrays. Simulation study shows that the proposed designs perform well compared with other sliced designs for computer experiments.

Sliced Latin hypercube designs via orthogonal arrays

Computer experiments are becoming increasingly popular in studying complex real world systems. A special class of sliced Latin hypercube design is proposed in this paper. Such designs are particularly suitable for computer experiments with both qualitative and quantitative factors, multi-fidelity computer experiments, cross-validation and data pooling. The resulting sliced Latin hypercube designs possess a desirable sliced structure and have an attractive low-dimensional uniformity. Meanwhile within each slice, it is also a Latin hypercube design with the same low-dimensional uniformity. The new sliced Latin hypercube designs can be constructed via both symmetric and asymmetric orthogonal arrays. The same desirable properties are possessed, although the uniformity may be differed. The construction methods are easy to implement, and unlike the existing methods, the resulting designs are very flexible in run sizes and numbers of factors. A detailed comparison with existing designs is made.

Asymmetrical Orthogonal Arrays With Run Size 100

Nowadays orthogonal arrays play important roles in statistics and other fields. Usual difference matrices are essential for the construction of many symmetrical or a few asymmetrical orthogonal arrays. But there are also asymmetrical orthogonal arrays which can not be obtained by the usual difference matrices. In order to construct these asymmetrical orthogonal arrays, a class of special matrices were discovered from the orthogonal decompositions of projection matrices. In this article, an interesting equivalent relationship between orthogonal arrays and the generalized difference matrices is presented. As an application, a lot of new orthogonal arrays of run size 100 have been constructed.

A lower bound to the measure of optimality for main effect plans in the general asymmetric factorial experiments

A design d is called D-optimal if it maximizes det(M d ) and is called MS-optimal if it maximizes tr(M d ) and minimizes tr[(M d )2] among those which maximize tr(M d ), where M d stands for the information matrix produced from d under a given model. In this paper, we establish a lower bound for tr[(M d )2] with respect to a main effects model, where d is an s 1×s 2×···×s m levels asymmetric orthogonal array of strength at least 1. Nonisomorphic asymmetrical MS-optimal orthogonal arrays of strength 1 with N=6, 8 and 12 runs are also presented.

Orthogonal arrays obtained by repeating-column difference matrices

In this paper, by using the repeating-column difference matrices and orthogonal decompositions of projection matrices, we propose a new general approach to construct asymmetrical orthogonal arrays. As an application of the method, some new orthogonal arrays with run sizes 72 and 96 are constructed.

A Simple Method for Constructing Orthogonal Arrays By the Kronecker Sum

In this article, we propose a new general approach to constructing asymmetrical orthogonal arrays, namely the Kronecker sum. It is interesting since a lot of new mixed-level orthogonal arrays can be obtained by this method.