The Association Schemes Based On Dickson Matrices

Let Fqn be a finite field of qn elements with characteristic p,Fqn*is a multiplicative group composed of all non-zero elements of Fqn,where q is the power of p,and n is a positive integer.The matrix of the form(?)is called the n-order Dickson matrix on Fqn,where ai?Fqn,i=0,1,…,n-1.All n-order Dickson matrices on Fqn form a ring under matrix addition and multiplication,denoted by R={[a0]+[a1]A+…+[an-1]An-1|ai?Fqn,i=0,1,…,n-1},where(?)and En-1 is an(n-1)×(n-1)identity matrix.Let R1*={[ai]Ai| ai ? F*qn,i=0,1,…,n-1},define a transformation ?P,Q,X0 on the ring R to satisfy?P,Q,X0(X)=PXQ+X0,(?)X ?R,then the group G={?P,Q,X0|P,Q?R1*,X0?R} acts transitively on the set R,which induces an association scheme on R,and we call it an association scheme based on Dickson matrices,denoted by Rn.Let S={D?R|tD=D},R*={D?R|det(D)?0},define a transformation ?P,X0 on the set S to satisfy?P,X0(X)=tPXP+X0,(?)X?S,then the group G={?P,X0|P?R*,X0 ? S} acts transitively on the set S,which also induces an association scheme on S,and we call it an association scheme based on the Symmetry Dickson matrices,denoted by Bn.In this paper,we researched the properties of the above two types of association schemes Rn and Bn in some special cases,and determined the classes and the intersection numbers of the association schemes R2 and B2(whenp=2).