Non-symmetric association schemes of symmetric matrices

LetX n be the set ofn ×n symmetric matrices over a finite fieldFq, whereq is a power of an odd prime. ForS1,S2 eX n , we define (S1,S2) eR0 iffS1 =S2; (S1,S2) eR(r,ξ) iffS1 -S2 is congruent to $$\left( {\begin{array}{*{20}c} {I^{(r - 1)} } & {} & {} \\ {} & \xi & {} \\ {} & {} & {0^{(n - r)} } \\ \end{array} } \right),$$ where ξ = 1 orz,z being a fixed non-square element ofF q . ThenX n = (X n , {R0,R(r,ξ) | 1 ≤r ≤n, ξ = 1 orz}) is a non-symmetric association scheme of class 2n onX n . The parameters ofX n have been computed. And we also prove thatX n is commutative.