Relative T-design On Hamming Association Scheme

The concept of relative t-designs for Q-polynomial association schemes is due to Del-sarte(1977). In2010, Bannai gave the Fisher type lower bound of relative2e-designs for Q-polynomial association schemes, and pointed out that it is very important to find clear Fishertype lower bound of relative t-designs for Q-polynomial association schemes. In this paper,using the theory of association schemes and the properties of the vector space, we obtain theFisher type lower bounds of tight relative2-,4-and6-designs very explicitly for binary Ham-ming association schemes H(n,2). Then we discuss some examples of tight relative2-and4-designs on H(n,2) in this sense, in connection with the work of Woodall(1970), Enomoto-Ito-Noda(1977), and others. Moreover, we study the relation of the tight relative2-designon binary Hamming association scheme H(n,2) and2-(n, k, λ) combinatorial design. Theproblems discussed above are very important in algebra combination. It is not only to developthe theory of relative t-design on association schemes, but also to have important applicationto the field of Information theory, computer science and engineering.The paper is consist of six chapters and organized as follows.In Chapter1, we introduce the background of our problems and describe others’ works ont-design, especially Bannai’s work on the Fisher type lower bound for the relative2e-designwith respect to u0on Q-polynomial association schemes. Then we list our main results.In Chapter2，we give some basic concepts and the basic lemmas.In Chapter3, firstly, using the theory of association scheme and the properties of thevector space, we obtain Fisher type lower bound in a very explicit form for relative2-designin binary Hamming association scheme H(n,2), and then discuss the relation between tightrelative2-design and2-(n, k, λ) design. Finally, we discuss some examples of tight relative2-design on binary Hamming association scheme H(n,2).In Chapter4, we study the relation between the binary Hamming association schemeH(n,2) and the Johnson association scheme, and using the relation above, we obtain the dimension of L2(Xk). Moreover, using the theory of association schemes and the propertiesof the vector space, we get Fisher type lower bound in a very explicit form for relative4-design in binary Hamming association scheme H(n,2). Finally, we discuss some examplesof tight relative4-design on binary Hamming association scheme H(n,2).In Chapter5, using the relation between the binary Hamming association scheme H(n,2)and the Johnson association scheme, we obtain the dimension of L3(Xk). Then using the factthat adjacency algebra of association scheme is isomorphic to intersection algebra of associ-ation scheme, we get the dimension of L3(Xk∪Xl). Finally, using the theory of associationschemes and the properties of the vector space, we obtain Fisher type lower bound in a veryexplicit form for the relative6-design in binary Hamming association scheme H(n,2).In Chapter6, we list some problems which will be further studied.