| Orthogonal arrays are essential in statistics and they are used in computer science arid cryptography- In statistics, they are primarily used in designing experiments, which simply means that they are immensely important in all areas of human investigation: for example, industry, agriculture, quality control and product improvement. One of the reasons why orthogonal arrays are so popular in experimental design is that they can be used as orthogonal main-effect plans. Mixed-level orthogonal arrays have additional appeal and receive more and more attention because they have great flexibility, allowing for factors with different numbers of levels. Orthogonal arrays are not only useful but also beautiful. The mathematical theory is extremely beautiful. Orthogonal arrays are related to combinatorics, finite fields, geometry and error-correcting codes, etc. The definition of an orthogonal array is simple and natural, and we know many elegant constructions-yet there are at least as many unsolved problems. And how to construct particular orthogonal designs for practical uses remains an open question in most instances.By using matrix theory and the theory of finite fields, we study the applications of projection matrix, permutation matrix, difference matrix, Hadamard product, the generalized Hadarnard product, Kronecker product, Kronecker sum and the generalized Kronecker sum to construction of orthogonal arrays. Some new methods of construction of orthogonal arrays especially mixed-level orthogonal arrays are presented. These methods produce many new orthogonal arrays which are laid out by mean of a software MATLAB. As applications of these methods, some combinatorial properties of orthogonal arrays are studied. The main results are listed in the following: Method for constructing orthogonal arrays by orthogonal decomposition of projection matrices and matrix images of smaller orthogonal arrays are further investigated. As the result, a lot of orthogonal arrays of run sizes 108 with many 6-level columns are constructed, especially including one array with 11 6-level columns, however, the existing orthogonal array with the same parameters has 288 mns. Also, two classes of orthogonal arrays with 4n2-and 9n2- runs are constructed, respectively. And series of orthogonal arrays are obtained. In orthogonal decomposition of projection matrices, a class of projection matrices Ir TP Tq are often encountered. We present a general method for constructing orthogonal arrays related to this class of projection matrices in various cases,which makes the constructed arrays have higher saturated percents. As an application of orthogonal decomposition of projection matrices, the method of juxtaposition is generalized. And we provide a general approach to construction of mixed-level orthogonal arrays by using generalized Hadamard product of two columns of an orthogonal array based on orthogonal decomposition of projection matrix. Moreover, the generalized Hadamard product of two columns of an orthogonal array is extended to that of two orthogonal arrays, which makes the constructed arrays have more non-prime power level columns. By applying the generalized Hadamard product and difference matrix to orthogonal decomposition of projection matrices, further results on the orthogonal arrays obtained by the generalized Hadamard product are presented. Several classes of orthogonal arrays having more columns than the existing arrays are obtained. By using the theory of finite fields, the relationship of module operation among difference matrices of a mixed difference matrix is generalized to that of group homomorphism. A new concept of normal mixed difference matrix is proposed. The relationship between normal mixed difference matrix and the construction of orthogonal arrays is discovered. A general method for constructing orthogonal arrays based on normal mixed difference matrix is presented. And some new orthogonal arrays are obtained. The remaining item of orthogonal decompositions of a projection matrix i...
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