Tops of graphs of projective codes

Let ${\mathrm{\Gamma }}_k(V)$ 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. Denote by $\mathrm{\Pi }[n,k{]}_q$ the subgraph of ${\mathrm{\Gamma }}_k(V)$ formed by projective codes. We give a complete description of cliques $⟨U{]}_k^{\mathrm{\Pi }}$ of $\mathrm{\Pi }[n,k{]}_q$ consisting of all $k$-dimensional projective codes contained in a fixed $(k+1)$-dimensional subspace of $V$. We show when and in how many lines of ${\mathcal{G}}_k(V)$ they are contained. Next we prove that $⟨U{]}_k^{\mathrm{\Pi }}$ is a maximal clique of $\mathrm{\Pi }[n,k{]}_q$ exactly if it is contained in at most one line of ${\mathcal{G}}_k(V)$.