arXiv 2603.040492026-03-04math.AGcs.ITmath.IT

微分戈普夫码

Differential Goppa Codes

Universidad de Salamanca（萨拉曼卡大学）；Universidad de León（莱昂大学）

David González González, Ángel Luis Muñoz Castañeda, Luis Manuel Navas Vicente

备注：: 43 pages, 0 figures

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AI总结：

本文研究了微分戈普夫码的构造与性质，通过n-jets理论和Hasse-Schmidt导数定义码字，并建立了对偶定理，证明戈普夫码是其真子类。

摘要：

罗森布洛姆和茨法曼在关于m-度量的基础工作中，引入了由平滑射影曲线X上的多个点定义的代数几何码。这种构造涉及一个除子G和另一个除子D=Σn p_i，其中p_i是不同的有理点，且p_i不在G的支撑集上，且n∈自然数。尽管这些码具有重要意义，但文献中对任意亏度的正式发展仍不完整，因为大多数研究都集中在亏度0的情况。我们对这类码进行了严谨的处理。从平滑射影曲线X、一个可逆sheaf L以及一个有效的除子D=Σn_i p_i开始，其中n_i不一定是相等的，以及在D的点上的uniformizers元组t_D和局部化L_{p_i}的trivializations γ_D，定义了相应的微分戈普夫码。这种码来源于曲线上的可逆sheaves的n-jets理论，这使得可以通过Hasse-Schmidt导数描述码字。在数据(t_D, γ_D)的变化下，码的变化被研究，并描述了作用在这组参数上的群。在这样的变化下，最小汉明距离的行为被分析，并为亏度0和1的曲线提供了具体的例子。建立了一个对偶定理，涉及单利部分的亚纯微分形式。证明了戈普夫码是微分戈普夫码的真子类，并且每个线性码都可以通过仅使用两个有理点在P^1上赋予微分戈普夫码结构。

英文摘要：

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.

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