Differential Goppa Codes
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arXiv
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| Formato: | Preprint |
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2026
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| _version_ | 1866908866001764352 |
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| author | González, David González Castañeda, Ángel Luis Muñoz Vicente, Luis Manuel Navas |
| author_facet | González, David González Castañeda, Ángel Luis Muñoz Vicente, Luis Manuel Navas |
| contents | Rosenbloom and Tsfasman, in their foundational work on the $m$-metric, introduced algebraic-geometric codes defined by multiple points on a smooth projective curve $X$. This construction involves a divisor $G$ and another divisor $D=\sum n p_i$, where $p_i$ are distinct rational points with $p_i \notin \text{supp}(G)$ and $n\in\mathbb{N}$. Although these codes are significant, their formal development for arbitrary genus remains incomplete in the literature, as most studies have concentrated on the genus $0$ case.
We present a rigorous treatment of this class of codes. Starting with a smooth projective curve $X$, an invertible sheaf $L$, and an effective divisor $D=\sum n_i p_i$ where the $n_i$ are not necessarily equal, as well as tuples of uniformizers $t_D$ at the points of $D$ and trivializations $γ_D$ for the localizations $L_{p_i}$, the associated differential Goppa code is defined. This code arises from the theory of $n$-jets of invertible sheaves on curves, which enables the description of codewords using Hasse-Schmidt derivatives of sections of $L$.
The variation of the code under changes in the data $(t_D, γ_D)$ is examined, and the group acting on these parameters is described. The behavior of the minimum Hamming distance under such variations is analyzed, with explicit examples provided for curves of genus $0$ and $1$. A duality theorem is established, involving principal parts of meromorphic differential forms. It is demonstrated that Goppa codes constitute a proper subclass of differential Goppa codes, and that every linear code admits a differential Goppa code structure on $\mathbb P^1$ using only two rational points. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_04049 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Differential Goppa Codes González, David González Castañeda, Ángel Luis Muñoz Vicente, Luis Manuel Navas Algebraic Geometry Information Theory 94B27, 94B05, 14B10, 13N15, 14G50 Rosenbloom and Tsfasman, in their foundational work on the $m$-metric, introduced algebraic-geometric codes defined by multiple points on a smooth projective curve $X$. This construction involves a divisor $G$ and another divisor $D=\sum n p_i$, where $p_i$ are distinct rational points with $p_i \notin \text{supp}(G)$ and $n\in\mathbb{N}$. Although these codes are significant, their formal development for arbitrary genus remains incomplete in the literature, as most studies have concentrated on the genus $0$ case. We present a rigorous treatment of this class of codes. Starting with a smooth projective curve $X$, an invertible sheaf $L$, and an effective divisor $D=\sum n_i p_i$ where the $n_i$ are not necessarily equal, as well as tuples of uniformizers $t_D$ at the points of $D$ and trivializations $γ_D$ for the localizations $L_{p_i}$, the associated differential Goppa code is defined. This code arises from the theory of $n$-jets of invertible sheaves on curves, which enables the description of codewords using Hasse-Schmidt derivatives of sections of $L$. The variation of the code under changes in the data $(t_D, γ_D)$ is examined, and the group acting on these parameters is described. The behavior of the minimum Hamming distance under such variations is analyzed, with explicit examples provided for curves of genus $0$ and $1$. A duality theorem is established, involving principal parts of meromorphic differential forms. It is demonstrated that Goppa codes constitute a proper subclass of differential Goppa codes, and that every linear code admits a differential Goppa code structure on $\mathbb P^1$ using only two rational points. |
| title | Differential Goppa Codes |
| topic | Algebraic Geometry Information Theory 94B27, 94B05, 14B10, 13N15, 14G50 |
| url | https://arxiv.org/abs/2603.04049 |