On the Grassmann Graph of Linear Codes

Let $\mathrm{\Gamma }(n,k)$ 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 ${\mathrm{\Delta }}_t(n,k)$ be the subgraph of $\mathrm{\Gamma }(n,k)$ formed by the set of linear $[n,k]$-codes having minimum dual distance at least $t+1$. We show that if $\mathrm{\mid }\mathbb{F}\mathrm{\mid }\ge \left(\genfrac{}{}{0px}{}{n}{t}\right)$ then ${\mathrm{\Delta }}_t(n,k)$ is connected and it is isometrically embedded in $\mathrm{\Gamma }(n,k)$. 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].