Tops of graphs of non-degenerate linear codes

Abstract

Let ${\Gamma }_k(V)$ be the Grassmann graph whose vertex set ${\mathcal{G}}_k(V)$ is formed by all $k$-dimensional subspaces of an $n$-dimensional vector space $V$ over the finite field $F_q$ consisting of $q$ elements. We discuss its subgraph $\Gamma (n,k{)}_q$ with the vertex set $\mathcal{C}(n,k{)}_q$ consisting of all non-degenerate linear $[n,k{]}_q$ codes. %We assume that $1<k<n-1$. We study maximal cliques $⟨U{]}_k^c$ of $\Gamma (n,k{)}_q$, which are intersections of tops of ${\Gamma }_k(V)$ with $\mathcal{C}(n,k{)}_q$. We show when they are contained in a line of ${\mathcal{G}}_k(V)$ and then we prove that $⟨U{]}_k^c$ is a maximal clique of $\Gamma (n,k{)}_q$ when it is not contained in a line of ${\mathcal{G}}_k(V)$. Furthermore, we show that the automorphism group of the set of such maximal cliques is isomorphic with the automorphism group of $\Gamma (n,k+1{)}_q$.