The graphs of non-degenerate linear codes

Abstract

We consider the Grassmann graph of $k$-dimensional subspaces of an $n$-dimensional vector space over the $q$-element field and its subgraph $\Gamma(n,k)_q$ 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 $\Gamma(n,k)_{q}$ 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 $\Gamma(n,k)_{q}$ and such that isomorphisms between these subgraphs and $\Gamma(n,k)_{q}$ cannot be extended to automorphisms of the Grassmann graph.