Constructions Of Mixed Orthogonal Arrays

Orthogonal array is a combinatorial structure introduced by the statistician C. R. Rao in1947when he studied experimental designs. It is also an important object in combinatorialdesigns and experimental designs. Many combinatorial mathematicians and statisticians de-voted themselves to the research of orthogonal arrays, and obtained abundant achievements.In addition, orthogonal array has close connection not only to many disciplines, such as statis-tics, computer science, coding cryptography, and so on, but also to many branches, such asfinite fields, number theory, finite geometry, experimental designs, etc.Many constructions and results of orthogonal array, further research problems and abun-dant of references are listed both in Orthogonal Arrays: Theory and Applications written byA. S. Hedayat, N. J. A. Sloane and J. Stufken in1999and in The CRC Handbook of Combi-natorial Designs written by R. J. R. Abel、C. J. Colbourn and J. H. Dinitz. The publicationof these books promotes the study of orthogonal array. With the development of informationscience and molecular biology, orthogonal array with various constraint conditions has beenput forward and investigated, such as mixed orthogonal arrays, mixed covering arrays, and soon. All these issues have important application backgrounds.In this paper, we mainly investigate the methods for constructions and the existence ofmixed orthogonal arrays and mixed covering arrays. The paper consists of five chapters andis organized as follows.In Chapter1, the research background of the full paper, the related concepts, the knownresults and our main results are presented.In Chapter2, we list some basic constructions of mixed orthogonal arrays and obtainthe existence of mixed orthogonal arrays with four and five factors of strength two (possiblyexcept one type) and the existence of mixed orthogonal arrays with t+1factors of strength t.In Chapter3, the methods for constructions of mixed orthogonal arrays with strengtht=3are investigated. We give three methods for constructing mixed orthogonal arrays, suchas difference matrix, Hadamard matrix and3-BD. As their applications, some new mixed orthogonal arrays with strength t=3are obtained.In Chapter4, the methods for constructions of covering arrays and mixed covering arraysare investigated. Firstly, we give the new upper bounds on some covering arrays with t=3, k=5, v≡2(mod4). Secondly, the existence of a class of optimal mixed covering arraysis given. Finally, we present some constructions of mixed covering arrays.In Chapter5, some problems for further research are listed.