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arXiv · 2509.17958

Tops of graphs of projective codes

Edyta Bartnicka
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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. Denote by $\Pi[n,k]_q$ the subgraph of $\Gamma_k(V)$ formed by projective codes. We give a complete description of cliques $\langle U]^{\Pi}_{k}$ of $\Pi[n,k]_q$ 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 $\langle U]^{\Pi}_{k}$ is a maximal clique of $\Pi[n,k]_q$ exactly if it is contained in at most one line of ${\mathcal G}_{k}(V)$.

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Edyta Bartnicka. 2025-09-22. Tops of graphs of projective codes. https://arxiv.org/abs/2509.17958

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