Recovering of the Grassmann graph from the subgraph of non-degenerate subspaces

Abstract

Let $\mathbb{F}$ be a (not necessarily finite) field. A subspace of the vector space ${\mathbb{F}}^n$ is called {\it non-degenerate} if it is not contained in a coordinate hyperplane. We show that the Grassmann graph of $k$-dimensional subspaces of ${\mathbb{F}}^n$, $1<k<n-1$, can be recovered from the subgraph of non-degenerate subspaces if $|\mathbb{F}|>n-k$. In the case when $\mathbb{F}={\mathbb{F}}_q$ is the field of $q$ elements, this subgraph is known as the graph of non-degenerate linear $[n,k{]}_q$ codes.