Tops of graphs of projective codes
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866912599633821696 |
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| author | Bartnicka, Edyta |
| author_facet | Bartnicka, Edyta |
| contents | 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. Denote by $Π[n,k]_q$ the subgraph of $Γ_k(V)$ formed by projective codes. We give a complete description of cliques $\langle U]^Π_{k}$ of $Π[n,k]_q$ 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 $\langle U]^Π_{k}$ is a maximal clique of $Π[n,k]_q$ exactly if it
is contained in at most one line of ${\mathcal G}_{k}(V)$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_17958 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Tops of graphs of projective codes Bartnicka, Edyta Combinatorics 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. Denote by $Π[n,k]_q$ the subgraph of $Γ_k(V)$ formed by projective codes. We give a complete description of cliques $\langle U]^Π_{k}$ of $Π[n,k]_q$ 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 $\langle U]^Π_{k}$ is a maximal clique of $Π[n,k]_q$ exactly if it is contained in at most one line of ${\mathcal G}_{k}(V)$. |
| title | Tops of graphs of projective codes |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2509.17958 |