Linear codes arising from the point-hyperplane geometry-Part I: the Segre embedding
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866908689341874176 |
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| author | Cardinali, Ilaria Giuzzi, Luca |
| author_facet | Cardinali, Ilaria Giuzzi, Luca |
| contents | Let $V$ be a vector space over the finite field $\mathbb{F}_q$ with $q$ elements and $Λ$ be the image of the Segre geometry $\mathrm{PG}(V)\otimes\mathrm{PG}(V^*)$ in $\mathrm{PG}(V\otimes V^*)$. Consider the subvariety $Λ_{1}$ of $Λ$ represented by the pure tensors $x\otimes ξ$ with $x\in V$ and $ξ\in V^*$ such that $ξ(x)=0$. Regarding $Λ_1$ as a projective system of $\mathrm{PG}(V\otimes V^*)$, we study the linear code $\mathcal{C}(Λ_1)$ arising from it. The code $\mathcal{C}(Λ_1)$ is minimal code and we determine its basic parameters, itsfull weight list and its linear automorphism group. We also give a geometrical characterization of its minimum and second lowest weight codewords as well as of some of the words of maximum weight. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_21309 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Linear codes arising from the point-hyperplane geometry-Part I: the Segre embedding Cardinali, Ilaria Giuzzi, Luca Combinatorics Discrete Mathematics Information Theory 51E22, 94B05, 14M12 Let $V$ be a vector space over the finite field $\mathbb{F}_q$ with $q$ elements and $Λ$ be the image of the Segre geometry $\mathrm{PG}(V)\otimes\mathrm{PG}(V^*)$ in $\mathrm{PG}(V\otimes V^*)$. Consider the subvariety $Λ_{1}$ of $Λ$ represented by the pure tensors $x\otimes ξ$ with $x\in V$ and $ξ\in V^*$ such that $ξ(x)=0$. Regarding $Λ_1$ as a projective system of $\mathrm{PG}(V\otimes V^*)$, we study the linear code $\mathcal{C}(Λ_1)$ arising from it. The code $\mathcal{C}(Λ_1)$ is minimal code and we determine its basic parameters, itsfull weight list and its linear automorphism group. We also give a geometrical characterization of its minimum and second lowest weight codewords as well as of some of the words of maximum weight. |
| title | Linear codes arising from the point-hyperplane geometry-Part I: the Segre embedding |
| topic | Combinatorics Discrete Mathematics Information Theory 51E22, 94B05, 14M12 |
| url | https://arxiv.org/abs/2506.21309 |