  Let $\Gamma(n,k)$ be the Grassmann graph formed by the $k$-dimensional
subspaces of a vector space of dimension $n$ over a field $\mathbb F$ and, for
$t\in \mathbb{N}\setminus \{0\}$, let $\Delta_t(n,k)$ be the subgraph of
$\Gamma(n,k)$ formed by the set of linear $[n,k]$-codes having minimum dual
distance at least $t+1$. We show that if $|{\mathbb F}|\geq{n\choose t}$ then
$\Delta_t(n,k)$ is connected and it is isometrically embedded in $\Gamma(n,k)$.
This generalizes some results of [M. Kwiatkowski, M. Pankov, "On the distance
between linear codes", Finite Fields Appl. 39 (2016), 251--263] and [M.
Kwiatkowski, M. Pankov, A. Pasini, "The graphs of projective codes" Finite
Fields Appl. 54 (2018), 15--29].
