Grassmannians of codes

Consider the point line-geometry ${\mathcal P}_t(n,k)$ having as points all the $[n,k]$-linear codes having minimum dual distance at least $t+1$ and where two points $X$ and $Y$ are collinear whenever $X\cap Y$ is a $[n,k-1]$-linear code having minimum dual distance at least $t+1$. We are interested in the collinearity graph $Λ_t(n,k)$ of ${\mathcal P}_t(n,k).$ The graph $Λ_t(n,k)$ is a subgraph of the Grassmann graph and also a subgraph of the graph $Δ_t(n,k)$ of the linear codes having minimum dual distance at least $t+1$ introduced in~[M. Kwiatkowski, M. Pankov, On the distance between linear codes, Finite Fields Appl. 39 (2016), 251--263, doi:10.1016/j.ffa.2016.02.004, arXiv:1506.00215]. We shall study the structure of $Λ_t(n,k)$ in relation to that of $Δ_t(n,k)$ and we will characterize the set of its isolated vertices. We will then focus on $Λ_1(n,k)$ and $Λ_2(n,k)$ providing necessary and sufficient conditions for them to be connected.