Subspaces in dd-bounded distance-regular graphs and their applications

Let ΓΓ be a dd-bounded distance-regular graph with diameter d≥3d≥3. For x∈V(Γ)x∈V(Γ), let P(x)P(x) be the set of all subspaces containing xx in ΓΓ. Suppose that 0≤t≤i+t,j+t≤i+j+t≤d1≤d0≤t≤i+t,j+t≤i+j+t≤d1≤d, and suppose that ΔΔ and Δ∗Δ∗ are subspaces with diameter i+ti+t and diameter d1d1 in P(x)P(x), respectively. Let Δ⊆Δ∗Δ⊆Δ∗; we give the number of subspaces Δ′Δ′ with diameter j+tj+t and Δ′⊆Δ∗Δ′⊆Δ∗ in P(x)P(x) such that d(Δ∩Δ′)=td(Δ∩Δ′)=t and d(Δ+Δ′)=i+j+td(Δ+Δ′)=i+j+t. Using the subspaces in P(x)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.