Tops of graphs of non-degenerate linear codes
Let $Γ_k(V)$ 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 $Γ(n,k)_q$ 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 $\langle U]^{c}_{k}$ of $Γ(n,k)_q$, which are intersections of tops of $Γ_k(V)$ 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 $\langle U]^{c}_{k}$ is a maximal clique of $Γ(n,k)_q$ 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 $Γ(n,k+1)_{q}$.