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 $|\mathbb{F}|\ge \left(\begin{array}{c}n\\ t\end{array}\right)$ then ${\mathrm{\Delta }}_t(n,k)$ is connected and it is isometrically embedded in $\mathrm{\Gamma }(n,k)$.

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关于线性码的格拉斯曼图

让 $\mathrm{\Gamma }(n,克)$是由一个域上的n维向量空间的k维子空间形成的 Grassmann 图$\mathbb{F}$ 并且，对于 $吨\in \mathbb{N}\setminus \{0\}$， 让 ${\mathrm{\Delta }}_{吨}(n,克)$ 是的子图 $\mathrm{\Gamma }(n,克)$ 由一组线性 $[n,克]$- 至少具有最小双距离的代码 $吨+1$. 我们证明如果$|\mathbb{F}|\ge \left(\begin{array}{c}n\\ 吨\end{array}\right)$ 然后 ${\mathrm{\Delta }}_{吨}(n,克)$ 是连接的，它等距地嵌入 $\mathrm{\Gamma }(n,克)$.