The Symplectic Fission Schemes For Association Schemes Of Rectangle Matrices

Let Fq be the finite field with q elements,Xm,n be the set of m x n matrices over Fq.Let GLm(Fq)denotes the general linear group of degree m over Fq,Sp2v(Fq)be the constructed group on Fq in terms of the symplectic matrix of 2vx2v nonsingular alternate matrix K versus matrix multiplication.Let G ={TA,T,Mo｜A∈GLm(Fq),M,Mo∈X,T ∈Sp2v(Fq)},we define the function of group G on set X as followsψ:GxXm,n→ Xm,n(TA,T,Mo,M)→ AMT+Mo.As the function of group G on Xm,n is transitive,thus it naturally induces a symmet-rical association scheme Xm,n,where A0,A1,…,Ad are the d + 1 orbits on Xm,n We call it the symplectic fission schemes of the rectangle matrix association schemes.This paper mainly computes the Xl,2v,X2,2v intersection numbers and determines the group of automorphism.