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Consider the Grassmann graph of $k$-dimensional subspaces of an$n$-dimensional vector space over the $q$-element field, $1<k<n-1$. Everyautomorphism of this graph is induced by a semilinear automorphism of thecorresponding vector space or a semilinear isomorphism to the dual vectorspace; the second possibility is realized only for $n=2k$. Let $\Gamma(n,k)_q$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$\Gamma(n,k)_{q}$ to a subgraph of the Grassmann graph can be uniquely extendedto an automorphism of the Grassmann graph. For $q=k=2$ there is an isomorphismof $\Gamma(n,k)_{q}$ to a subgraph of the Grassmann graph which does not havethis property. In this paper, we show that such exceptional isomorphism isunique up to an automorphism of the Grassmann graph.

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