论文标题
非分类线性代码的图
The graphs of non-degenerate linear codes
论文作者
论文摘要
我们考虑了$ n $二维矢量空间的$ k $维二维子空间的Grassmann图，$ q $ element字段及其子图$γ（N，K）_Q $由非分数线性$ [N，K] _Q $代码形成。我们假设$ 1 <k <n-1 $。众所周知，格拉斯曼图的每一个自动形态都是由相应的矢量空间的半线性自动形态或对双矢量空间的半线性同构的；仅当$ n = 2k $时，才能实现第二种可能性。我们的结果如下：如果$ q \ ge 3 $或$ k \ ne 2 $，那么每一个$γ（n，k）_ {q} $的同构为Grassmann图的一个子图，可以独特地扩展到Grassmann图的自动形态上；在$ q = k = 2 $的情况下，格拉斯曼图同构的子图至$γ（n，k）_ {q} $，因此这些子毛学和$γ（n，k）_ {q} $之间的同构性不能扩展到Grassmann图的自动形态。
We consider the Grassmann graph of $k$-dimensional subspaces of an $n$-dimensional vector space over the $q$-element field and its subgraph $Γ(n,k)_q$ formed by non-degenerate linear $[n,k]_q$ codes. We assume that $1<k<n-1$. It is well-known that every automorphism of the Grassmann graph is induced by a semilinear automorphism of the corresponding vector space or a semilinear isomorphism to the dual vector space; the second possibility is realized only if $n=2k$. Our results are the following: if $q\ge 3$ or $k\ne 2$, then every isomorphism of $Γ(n,k)_{q}$ to a subgraph of the Grassmann graph can be uniquely extended to an automorphism of the Grassmann graph; in the case when $q=k=2$, there are subgraphs of the Grassmann graph isomorphic to $Γ(n,k)_{q}$ and such that isomorphisms between these subgraphs and $Γ(n,k)_{q}$ cannot be extended to automorphisms of the Grassmann graph.
