The Association Schemes And The Schematic Orthogonal Arrays Obtained From Orthogonal Arrays Constructed By Difference Matrix

Orthogonal arrays are very useful. They are not only used in statistics, but also incoding theory, computer science, cryptography, etc. Association scheme is a combinatorialstructure in balance incomplete block design and it describes the balance of symbols withmany associate relations. With the further study of association schemes, it has been foundthat association schemes have close relations with group, graph theory, coding theory andcombinatorics. The study of association scheme has developed into an important branch ofalgebraic combinatorics. In the theory of orthogonal arrays, an orthogonal array is calledschematic if its rows form an association scheme with respect to Hamming distances.This paper extends the definitions of Hamming distance and schematic orthogonal ar-ray to allow the columns to have diferent levels. We then study the Hamming distances ofthe rows in a class of orthogonal arrays obtained by diference matrix, A general construc-tion method of association schemes according to the distances is presented and some of theorthogonal arrays are proved to be schematic. Some association schemes and schematicorthogonal arrays are provided to illustrate the applications of the method.Chapter1introduces the development and the current research status of the orthog-onal array and association scheme, as well as the basic definitions of orthogonal array andmain lemmas.Chapter2mainly extends the definition of Hamming distance of any two rows of or-thogonal array and presents a new concept of schematic mixed orthogonal arrays. Then westudy the Hamming distances of the rows in orthogonal arrays(symmetric or asymmetric)constructed by special diference matrix(the transpose of diference matrix is also a difer-ence matrix), some constructing methods of association schemes of orthogonal arrays aregiven and they are further proven that they are schematic orthogonal arrays under someconditions. At last, Some examples are given to illustrate these methods of constructingschematic orthogonal arrays and association schemes.Chapter3concludes the main content of this paper and puts forward some suggestions and a few unsolved problems.