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.

翻译：Rosenbloom与Tsfasman在其关于$m$-度量的奠基性工作中，引入了由光滑射影曲线$X$上多点定义的代数几何码。该构造涉及一个除子$G$与另一个除子$D=\sum n p_i$，其中$p_i$为互异的有理点且满足$p_i \notin \text{supp}(G)$，$n\in\mathbb{N}$。尽管这类码具有重要意义，但针对任意亏格情形的形式化发展在文献中仍不完整，现有研究多集中于亏格$0$的情形。本文对此类码进行了严格处理。从一条光滑射影曲线$X$、一个可逆层$L$以及一个有效除子$D=\sum n_i p_i$出发（其中$n_i$未必相等），并给定$D$点处的一致化参数组$t_D$与局部化$L_{p_i}$的平凡化映射$γ_D$，可定义相应的微分Goppa码。该码源于曲线上可逆层的$n$-jet理论，使得码字可通过$L$截面的Hasse-Schmidt导数进行描述。本文考察了码在参数组$(t_D, γ_D)$变化下的演变，刻画了作用于这些参数上的群结构，并分析了此类变化下最小汉明距离的行为特性，同时给出了亏格$0$与$1$曲线的具体算例。文中建立了一个对偶定理，该定理涉及亚纯微分形式的主部。研究证明：Goppa码构成微分Goppa码的真子类，且任意线性码均可通过$\mathbb P^1$上仅两个有理点实现为微分Goppa码结构。