The S-symplectic Fission Scheme For Association Schemes Of Rectangle Matrices And Its Applications

Let Fqbe a finite field with q elements,Mm,2v+l(Fq)denotes the set of all m ×(2v+l)matrices over Fq,which forms an Abelian group under matrix addition.Sp2v+l,v(Fq)is the singular symplectic group of degree 2v +l and index v over Fq,GLm(Fq)is the general linear group of degree m over Fq.Let G0 = GLm(Fq)× Sp2v+l,v(Fq),G0 acts on Mm,2v+/(Fq)as follows:Mm,2v+l(Fq)× G0→Mm,2v+l(Fq)(M,(P,Q))→PMQ.Obviously,(P,Q)gives an automorphism for the addition group Mm,2v+l(Fq),so G0 is an automorphism group of Mm,2v+l and induces an association scheme on Mm,2v+l,we call it the S-singular fission scheme for association schemes of rectangle matrices which is denoted by mm,2v+l,v.Let X be a finite Abelian group,G0 be an automorphism group of X.X0 ={0},X1,…Xd,Xd are all G0 orbits of X,then G0 induces an association scheme on X wihch is denoted by X.F is a field,r,c,k are integers where 0 ≤ r,c,k<d.Let m = |Xr|,n = |Xc|,define an m × n matrix over F whose rows and columns are indexed by the elements of Xr and Xc respectively as follows:Arck(x,y)= 1,y-x∈Xk,0,otherwise.Arck is called the matrix of x over F with parameters r,c,k.In this paper,we research the association schemes mm,2v+l,v and construct a class of linear codes,the main results are:(1)The formulas for class numbers,valences and some intersection numbers of mm,2v+l,v;(2)All intersection numbers of mm,2v+l,v when m = 1,2;(3)Constructing a class of linear codes by using mm,2v+l,v and the matrix definited above,and discussing their parameters.