The Association Schemes Based On Square Matrices Over Finite Fields

Let F_q is a finite field of q elements,F_q*is a cyclic group of order q-1 with respect to field multiplication,α is the generator of F_q*,Aut(F_q)is the automorphism group of F_q.The set of n x n nonsingular matrices over F_q forms a group under matrix multiplication,called the general linear group of degree n over F_q and denoted by GLn(F_q).Let X is a set of all n x n matrix over finite fields,the group of all of translation over F_q denoted by T(X).All of the following forms of transformation over XA→kP-1 AτP 其中k∈F_q*,P∈GLn(F_q),A∈X,τ∈Aut(F_q)(2)is an automorphism of add group X denoted by σ(k,P,τ),the set of such automorphism forms a group under map multiplication denoted by G0.Let G be a group generated by G0 and T(X),the action of the group G on X,is transitive,and induces naturally an association scheme denoted by (?)n.This association scheme is a fission scheme of rectangular matrix.In this dissertation,the normalized forms and a part of intersection numbers of association scheme (?)2 are determined.