Double covers of symplectic dual polar graphs

Let Γ=Γ(2n,q)Γ=Γ(2n,q) be the dual polar graph of type Sp(2n,q)Sp(2n,q). Underlying this graph is a 2n2n-dimensional vector space VV over a field FqFq of odd order  qq, together with a symplectic (i.e. nondegenerate alternating bilinear) form B:V×V→FqB:V×V→Fq. The vertex set of ΓΓ is the set VV of all nn-dimensional totally isotropic subspaces of  VV. If q≡1q≡1 mod 4, we obtain from ΓΓ a nontrivial two-graph Δ=Δ(2n,q)Δ=Δ(2n,q) on  VV invariant under PSp(2n,q)PSp(2n,q). This two-graph corresponds to a double cover Γ̂→Γ on which is naturally defined a QQ-polynomial (2n+1)(2n+1)-class association scheme on 2|V̂| vertices.