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

TL;DR

This paper demonstrates that the Grassmann graph of certain subspaces can be reconstructed from a subgraph of non-degenerate subspaces, especially over finite fields, with implications for coding theory.

Contribution

It establishes conditions under which the Grassmann graph can be recovered from the non-degenerate subgraph, extending understanding in vector space and coding theory.

Findings

Grassmann graph can be reconstructed from non-degenerate subgraph when |F| > n-k

The subgraph corresponds to the graph of non-degenerate linear codes over finite fields

Recovery is possible for 1<k<n-1 in vector spaces over arbitrary fields

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.