Tops of graphs of non-degenerate linear codes
Edyta Bartnicka, Andrzej Matraś
TL;DR
This work analyzes the subgraph $\Gamma(n,k)_q$ of the Grassmann graph restricted to non-degenerate linear codes ${\mathcal C}(n,k)_q$, focusing on maximal cliques that arise as intersections of tops with the non-degenerate set. It introduces a key construction using a subspace $W\subset V^{k+1}$ generated by vectors not proportional to columns of a fixed generator matrix, and shows that $\langle U]^{c}_{k}$ lies in a line precisely when $\dim W\le 2$; otherwise, $\langle U]^{c}_{k}$ is a maximal clique (a top) of $\Gamma(n,k)_q$. The paper then identifies the automorphism group of the set of such tops with $Aut(\Gamma(n,k+1)_q)$, showing that automorphisms are induced by monomial semilinear automorphisms of $V$ (with explicit considerations for primes and for $q=2$). Overall, it provides a complete description of when tops lie on lines, proves that non-line-contained tops are maximal cliques, and links the automorphism structure of these tops to the graph of non-degenerate codes in one higher dimension, advancing code equivalence perspectives via a graph-theoretic lens.
Abstract
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 $Γ(n,k+1)_{q}$.