On the Grassmann graph of linear codes

Let Γ(n, k) be the Grassmann graph formed by the k-
dimensional subspaces of a vector space of dimension n over a
field F and, for t ∈ N {0}, let Δ t (n, k) be the subgraph
of Γ(n, k) formed by the set of linear [n, k]-codes having
minimum dual distance at least t +1. We show that if |F| ≥ nt
then Δ t (n, k) is connected and it is isometrically embedded
in Γ(n, k).