Spherical 5-designs obtained from finite unitary groups

Let G=U_2m(2) be the unitary group of dimension 2m≥6 over the finite field of four elements GF(4), W=GF(4)^2m the natural module of G. Then G acts transitively on the set Ω of maximal totally isotropic m-dimensional subspaces of W. This permutation representation over R contains an irreducible representation of dimension d=(4^m+2)/3. One can embed the set Ω into the unit sphere S^d-1 in the Euclidean space R^d, and we prove that this embedding gives a spherical 5-design.