Association Schemes Obtained From A Class Of Orthogonal Arrays By Deleting Some Columns

Association scheme is a part of what is now called algebraic combinatorics, which was defined by R.C. Bose and T.Shimamoto in1952, it is a combinatorics structure in balance incomplete block designs and it describes the balance of symbols with many associate relations. With the further study of association scheme, it was found to have close relations with group, coding theory, graph theory and combinatorics. The study of association scheme has developed into an important branch of algebraic combinatorics. Especially it provides a theoretical framework for the coding theory, so the study of association scheme has developed into an important branch of algebraic combinatorics.In the theory of orthogonal arrays, an orthogonal array is called 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 Application》. One of these 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 strength2. In this paper we study the Hamming distances of the rows in orthogonal arrays and give the schematic orthogonal arrays with strength2and the construction of association schemes.Chapter1introduces the development and the current research status of orthogonal arrays and association scheme, and contains basic concepts and main lemmas.Chapter2extends 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 deleting 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 is presented and some of the orthogonal arrays are proved to be schematic. Some association schemes and schematic orthogonal arrays are provided to illustrate the applications of the method. Chapter3concludes the main content of this paper and puts forward some suggestionsand a few unsolved problems.