Schematic Orthogonal Arrays Obtained By Using Expansive Replacement Method

Association scheme is widely researched and applied, which was defined by R.C. Bose and T.Shimamoto in 1952, it is gradually applied to coding theory, statistics and computer science, etc. The study of association scheme has developed into an important branch of algebraic combinatorics,it reaches the balance of symbols among many associate rela-tions. Therefore research association scheme has become an important issue.In the theory of orthogonal arrays, an orthogonal array is schematic if its rows form an association scheme with respect to Hamming distances. A.S. Hedayat studied the orthogonal arrays in detail and proposed many open problems in (《Orthogonal Arrys:Theory and Applicationed》. One of the important problems is that in which orthogonal arrays the rows form an association scheme according to Hamming distances and how to classify them. At present, it is very few about the results of orthogonal arrays of association scheme. However, there are very few researches on the orthogonal arrays of the mixed. In this paper we study the Hamming distances of the rows and the expansive replacement method in orthogonal arrays and give the schematic mixed orthogonal arrays and the construction of association schemes.In chapter l,we introduce the development and the current research status of orthog-onal arrays and association scheme, and contains basic concepts and main lemmas.In chapter 2,we extend the definitions of Hamming distance and schematic orthogonal array to allow the columns to have different levels, we then study the Hamming distances of the rows in a class of orthogonal arrays obtained by replacing one or two columns from orthogonal arrays which are related with a class of difference matrix. A general construction method of association schemes according to the distances and the arrays are presented and some of the orthogonal arrays are proved to be schematic. Some association schemes and schematic orthogonal arrays are provided to confirm the applications of the method.