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 $\Lambda_t(n,k)$ of ${\mathcal P}_t(n,k).$ The graph $\Lambda_t(n,k)$ is a subgraph of the Grassmann graph and also a subgraph of the graph $\Delta_t(n,k)$ 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 $\Lambda_t(n,k)$ in relation to that of $\Delta_t(n,k)$ and we will characterize the set of its isolated vertices. We will then focus on $\Lambda_1(n,k)$ and $\Lambda_2(n,k)$ providing necessary and sufficient conditions for them to be connected.

翻译：考虑点线几何${\mathcal P}_t(n,k)$，其中点集为所有最小对偶距离至少为$t+1$的$[n,k]$线性码，且两点$X$和$Y$共线当且仅当$X\cap Y$是一个最小对偶距离至少为$t+1$的$[n,k-1]$线性码。本文研究${\mathcal P}_t(n,k)$的共线图$\Lambda_t(n,k)$。图$\Lambda_t(n,k)$既是格拉斯曼图的子图，也是文献[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]中引入的最小对偶距离至少为$t+1$的线性码图$\Delta_t(n,k)$的子图。我们将结合$\Delta_t(n,k)$的结构研究$\Lambda_t(n,k)$的性质，并刻画其孤立顶点集。随后重点分析$\Lambda_1(n,k)$和$\Lambda_2(n,k)$，给出它们连通性的充要条件。