Dual of Algebraic Geometry codes from Hirzebruch surfaces
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866914410773086208 |
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| author | Barraud, Alix |
| author_facet | Barraud, Alix |
| contents | In this paper, we give an explicit form for the dual of the algebraic geometry code $C_e(a,b)$ defined on an Hirzebruch surface $\mathcal{H}_e$ and parametrized by the divisor $aS_e + bF_e$, where $a,b\in\mathbb{N}$ and $S_e$ and $F_e$ generate the Picard group $\mathrm{Pic}( \mathcal{H}_e)$. Notably, we compute a lower bound for the minimum distance of $C_e(a,b)^\perp$. One of the main ingredient for our study is a new explicit form of the code $C_e(a,b)$ which we provide at the beginning of the paper. We also investigate some puncturing of $C_e(a,b)$, recovering other previously studied AG codes from toric surfaces. Finally, we provide a sufficient condition for orthogonal inclusions between the codes $C_e(a,b)$, and construct CSS quantum codes from them. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_07761 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Dual of Algebraic Geometry codes from Hirzebruch surfaces Barraud, Alix Algebraic Geometry Information Theory 94B27, 14J26, 11G25, 14C20 In this paper, we give an explicit form for the dual of the algebraic geometry code $C_e(a,b)$ defined on an Hirzebruch surface $\mathcal{H}_e$ and parametrized by the divisor $aS_e + bF_e$, where $a,b\in\mathbb{N}$ and $S_e$ and $F_e$ generate the Picard group $\mathrm{Pic}( \mathcal{H}_e)$. Notably, we compute a lower bound for the minimum distance of $C_e(a,b)^\perp$. One of the main ingredient for our study is a new explicit form of the code $C_e(a,b)$ which we provide at the beginning of the paper. We also investigate some puncturing of $C_e(a,b)$, recovering other previously studied AG codes from toric surfaces. Finally, we provide a sufficient condition for orthogonal inclusions between the codes $C_e(a,b)$, and construct CSS quantum codes from them. |
| title | Dual of Algebraic Geometry codes from Hirzebruch surfaces |
| topic | Algebraic Geometry Information Theory 94B27, 14J26, 11G25, 14C20 |
| url | https://arxiv.org/abs/2509.07761 |