A Characterization of the Association Schemes of Bilinear Forms

If n ⩾ 2d ⩾ 6 and q ⩾ 4, the distance-regular graphs Hq (n, d) (with d × n matrices over GF(q) as vertex set, and two vertices A, B being adjacent if and only if the rank of A − B is 1) are characterized by their parameters together with the property that the number of edges of the induced subgraph on the common neighborhood of vertices x and y depends only on the distance between them. As a corollary, we prove that Hq (n, d) has no antipodal covering if n ⩾ 2d ⩾ 6 and q ⩾ 4