The Orthogonal Fission Scheme For Association Schemes Of Rectangle Matrices Over The Finite Fileds Of Characteristic 2 And Its Application

Let Fq is a finite field of characteristic 2.We assume that Xm,n,is the set of all m ×n matrices over Fq.GLm(Fq)denotes the general linear group of degree m over Fq.And On(Fq,G)is consisted of all orthogonal matrices with respect to the regular matrix G of degree n over Fq,called the orthogonal group.The transformation ?A,T,M0 acts on Xm,n as follows:M(?)AMT+M0,(?)M ?Xm,n,where A ? Gm(Fq),T ? On(Fq,G),and M0 ? Xm,n.The group made up of such transformation is denoted by H.The group H acts transitively on the set Xm,n,and naturally induces an association scheme on Xm,n,which is denoted by(?)m,n=(Xm,n,{Ri}0?i?d),of which R0,R1,...,Rd are d+1 orbits induced by the group H acts on Xm,n × Xm,n.It is a fission scheme for the association scheme of rectangle matrices,called the orthogonal fission schemes for association schemes of rectangle matrices.Following the association scheme(?)m,n constructed,this paper calculates intersection numbers of the association scheme when m=1 and when m=2 first,and then constructs a class of binary linear code by using the association scheme(?)m,n.The upper bound of the minimum distance of the linear code in some particular cases is also given.