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

TL;DR

This paper investigates the automorphisms of the subgraph of the Grassmann graph formed by non-degenerate linear codes, establishing conditions under which these automorphisms extend uniquely to the entire Grassmann graph, with special cases analyzed.

Contribution

The paper characterizes when automorphisms of the non-degenerate code subgraph extend uniquely to the Grassmann graph, highlighting a unique exceptional isomorphism for the case q=k=2.

Findings

Automorphisms extend uniquely for q≥3 or k≥3.

Exceptional isomorphism exists for q=k=2.

Such exceptional isomorphism is unique up to automorphism.

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 $\Gamma(n,k)_q$ 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 extended to an automorphism of the Grassmann graph. For $q=k=2$ there is an isomorphism of $\Gamma(n,k)_{q}$ 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.