Subspaces in -bounded distance-regular graphs and their applications

Let Γ be a d -bounded distance-regular graph with diameter d ≥ 3 . For x ∈ V ( Γ ) , let P ( x ) be the set of all subspaces containing x in Γ . Suppose that 0 ≤ t ≤ i + t , j + t ≤ i + j + t ≤ d 1 ≤ d , and suppose that Δ and Δ ∗ are subspaces with diameter i + t and diameter d 1 in P ( x ) , respectively. Let Δ ⊆ Δ ∗ ; we give the number of subspaces Δ ′ with diameter j + t and Δ ′ ⊆ Δ ∗ in P ( x ) such that d ( Δ ∩ Δ ′ ) = t and d ( Δ + Δ ′ ) = i + j + t . Using the subspaces in P ( x ) , we construct a new Cartesian authentication code. We also compute its size parameters and its probabilities of successful impersonation attack and of successful substitution attack.