On minimal codes arising from projective embeddings of point-line geometries
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
| Veröffentlicht: |
2025
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| _version_ | 1866917461789507584 |
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| author | Cardinali, Ilaria Giuzzi, Luca |
| author_facet | Cardinali, Ilaria Giuzzi, Luca |
| contents | Let ${\mathcal C}(Ω)$ be the linear code arising from a projective system $Ω$ of $\mathrm{PG}(V).$ Consider the point-line geometry $Γ=({\mathcal P},{\mathcal L})$ and a projective embedding $\varepsilon\colon Γ\rightarrow \mathrm{PG}(V)$ of $Γ.$ We show that the projective code obtained by taking as projective system $Ω:=\varepsilon(\mathcal{P})$ is minimal if the graph induced on the set $Γ\setminus\varepsilon^{-1}(H)$ by the collinearity graph of $Γ$ is connected for any hyperplane $H$ of $\mathrm{PG}(V)$. As an application, Grassmann codes, Segre codes, polar Grassmann codes of orthogonal, symplectic, hermitian type and codes arising from the point-hyperplane geometry of a projective space are minimal codes. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2511_22747 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On minimal codes arising from projective embeddings of point-line geometries Cardinali, Ilaria Giuzzi, Luca Combinatorics Information Theory 51E22, 51A50, 14M15 Let ${\mathcal C}(Ω)$ be the linear code arising from a projective system $Ω$ of $\mathrm{PG}(V).$ Consider the point-line geometry $Γ=({\mathcal P},{\mathcal L})$ and a projective embedding $\varepsilon\colon Γ\rightarrow \mathrm{PG}(V)$ of $Γ.$ We show that the projective code obtained by taking as projective system $Ω:=\varepsilon(\mathcal{P})$ is minimal if the graph induced on the set $Γ\setminus\varepsilon^{-1}(H)$ by the collinearity graph of $Γ$ is connected for any hyperplane $H$ of $\mathrm{PG}(V)$. As an application, Grassmann codes, Segre codes, polar Grassmann codes of orthogonal, symplectic, hermitian type and codes arising from the point-hyperplane geometry of a projective space are minimal codes. |
| title | On minimal codes arising from projective embeddings of point-line geometries |
| topic | Combinatorics Information Theory 51E22, 51A50, 14M15 |
| url | https://arxiv.org/abs/2511.22747 |