Chow's theorem for linear codes

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 $V$.