Schematic Orthogonal Arrays Obtained By Using Contractive Replacement Method

Orthogonal arrays have wide applications. They are essential in statistics and theyare used in coding theory, cryptography and computer science, etc. Association scheme isa combinatorics structure in balance incomplete block design, it describes the balance ofsymbols with many associate relations. With the further study of association scheme, it wasfound to have close relations with group, graph theory, coding theory and combinatorics.The study of association scheme has developed into an important branch of algebraiccombinatorics.In the theory of orthogonal arrays, an orthogonal array is called schematic if its rowsform an association scheme with respect to Hamming distances, which was defined byBose and Shimamoto in1952. In1973, its theory was used in designing and coding.The orthogonal arrays which forms the association scheme of two classes are importantin statistics. However, there are few researches on the orthogonal arrays of strength2.Therefore, Hedayat listed it as an open problem in “Orthogonal Arrays: Theory andApplication”.Chapter1introduces the development and the current research status of orthogonalarrays and association scheme, and contains basic concepts and main lemmas.Chapter2extends the definitions of Hamming distance and schematic mixed orthog-onal array, then we study the Hamming distances of the rows in asymmetric orthogonalarrays constructed by using Contractive replacement method, some constructing methodsof association schemes of these orthogonal arrays are given and they are further proven thatthey are schematic mixed orthogonal arrays. At last, Some examples are given to illustratethese methods of constructing schematic orthogonal arrays and association schemes.Chapter3concludes the main content of this paper and puts forward some suggestionsand a few unsolved problems.