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：考虑具有最小双重距离至少为$t+1$的所有$[n,k]$-线性码的点线几何图${\mathcal P}_t(n,k)$，其中当$X\cap Y$是一个最小双重距离至少为$t+1$的$[n,k-1]$-线性码时，两个点$X$和$Y$是共线的。我们对${\mathcal P}_t(n,k)$的共线性图$\Lambda_t(n,k)$感兴趣。图$\Lambda_t(n,k)$是Grassmann图的子图，也是线性码最小双重距离至少为$t+1$的图$\Delta_t(n,k)$的子图，该图在[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。我们将研究$\Lambda_t(n,k)$的结构与$\Delta_t(n,k)$的结构有关，并且将刻画其孤立点的集合。然后，我们将重点关注$\Lambda_1(n,k)$和$\Lambda_2(n,k)$，并提供它们连接的必要和充分条件。
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