Some Association Schemes Constructed From The Kronecker Sum Of The Orthogonal Arrays

Association scheme is a combinatorics structure in Partially Balanced Incomplete Block Designs, which describes the balance of symbols with many associate relations. it was found to have close relations with coding theory, design theory and finite group theory. The definition of orthogonal array is simple and natural, its mathematical theory involves combinatorial mathematics, limited domain, the geometry and the error correcting code, etc. What orthogonal arrays of strength2form association schemes according to the rela-tions of the rows and how to classify, Hedayat listed it as an open problem in "Orthogonal Arrays:Theory and Application".Making use of the orthogonal arrays already constructed, we study the generalized Hamming Distance (also called Hamming Distance for short) among the row vectors of each Orthogonal Array and extend the definition of schematic orthogonal array. According to the generalized Hamming Distance and classifying the row vectors of each orthogonal array,2-class and several classes association schemes are obtained and some schematic orthogonal arrays are found. Some specific examples are given to illustrate each theorem.This paper is divided into four chapters:Chapter1introduces the development and the current research status of association schemes and orthogonal arrays, and contains some basic concepts and main lemmas.Chapter2considers that according to the Hamming Distance and classifying the row vectors of the orthogonal array Ls2(sg) as two classes, the orthogonal array Ls2(sg) of strength2is equivalent to a Latin square scheme Lg(s) of two classes, and obtains a set of g-2mutually orthogonal latin squares of order s by Ls2(sg).Chapter3firstly discusses according to the Hamming Distance and classifying the row vectors of the orthogonal array as several classes, then analyzes the connection between two different classes having the same Hamming distances and their intersection matrixes, a necessary and sufficient condition, when two different classes could be merged into one class, is presented. Secondly, this necessary and sufficient condition is used in the middle of the following theorem, we can find some schematic Orthogonal Arrays. Finally, parametersof the association scheme constructed by the orthogonal arrays is very large, it takes up alot of lengths.Chapter4summarizes the results of this master’s thesis, put forward constructivesome comments and suggestions, as well as some outstanding issues.