On the graph of non-degenerate linear $[n,2]_2$ codes

Consider the Grassmann graph of $k$-dimensional subspaces of an $n$-dimensional vector space over the $q$-element field, $1<k<n-1$. Every automorphism of this 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 for $n=2k$. Let $Γ(n,k)_q$ be the subgraph of the Grassman graph formed by all non-degenerate linear $[n,k]_q$ codes. If $q\ge 3$ or $k\ge 3$, 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. For $q=k=2$ there is an isomorphism of $Γ(n,k)_{q}$ to a subgraph of the Grassmann graph which does not have this property. In this paper, we show that such exceptional isomorphism is unique up to an automorphism of the Grassmann graph.