Chow's theorem for linear codes

TL;DR

This paper characterizes the automorphisms of a graph formed by non-degenerate linear codes, showing they are induced by monomial semilinear automorphisms, and describes the maximal cliques within this graph.

Contribution

It provides a complete description of the automorphism group of the code-based Grassmann graph and characterizes its maximal cliques, extending the understanding of symmetries in linear code spaces.

Findings

Automorphisms are induced by monomial semilinear automorphisms.

Maximal cliques of the graph are characterized.

The structure of the Grassmann graph for linear codes is elucidated.

Abstract

Let $\Gamma_{k}(V)$ be the Grassmann graph formed by $k$-dimensional subspaces of an $n$-dimensional vector space over the finite field ${\mathbb F}_{q}$ consisting of $q$ elements and $1<k<n-1$. Denote by $\Gamma(n,k)_q$ the restriction of the Grassmann graph to the set of all non-degenerate linear $[n,k]_q$ codes. We describe maximal cliques of the graph $\Gamma(n,k)_q$ and show that every automorphism of this graph is induced by a monomial semilinear automorphism of $V$.