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

Abstract

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≥3 or k≥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.