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 $\langle U]^{c}{k}$ 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 $\langle U]^{c}_{k}$ 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 $\Gamma(n,k+1)_{q}$.