发表机构
United States Military Academy; Oakland University（西点军校; 奥克兰大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文研究加权超椭圆曲线上一类码的置换与单项式等价，证明其笛卡尔字符码结构，确定自同构群与最小距离，并给出壳维数界及不变量应用。
AI 中文摘要
我们研究了加权超椭圆曲线上一点码的置换等价与单项式等价。在$H_m\simeq \Z/n\Z\times\Z/m\Z$正则作用于求值集的层上，我们证明了这些码是笛卡尔字符码，并确定了它们的本原最小距离和对偶最小距离。我们计算了正则作用的射影单项式正规化子，并通过定义集上的诱导作用对相容的置换等价与单项式等价进行了分类。当$d > n$且$m \geq 2$时，完整滤过的公共置换自同构群恰好是$H_m$，且正则作用可由其两个最低的非恒定成员恢复。我们在滤过顶端以闭式形式确定了完整的置换自同构群，并给出了在所处理的其余本原情形中$\PAut ( \CC_s ) = H_m$的与域无关的有限证书。除类型$(3,3,3)$外，在每个正则层上，壳维数受亏格限制。显式例子表明，长度、维数和壳维数不能确定单项式等价，且壳轮廓与正则自同构子群一起不能确定曲线直至同构。作为应用，我们确定了支持分裂方法中使用的相应基于壳的不变量和穿孔签名。
英文摘要
We study permutation and monomial equivalence for one-point codes on weighted superelliptic curves. On strata where $H_m\simeq \Z/n\Z\times\Z/m\Z$ acts regularly on the evaluation set, we prove that these codes are Cartesian character codes and determine their primal and dual minimum distances. We compute the projective monomial normalizer of the regular action and classify the compatible permutation and monomial equivalences through the induced actions on the defining sets. When $d > n$ and $m \geq 2$, the common permutation automorphism group of the full filtration is exactly $H_m$, and the regular action can be recovered from its two lowest nonconstant members. We determine the full permutation automorphism group in closed form at the top of the filtration and give a field-independent finite certificate for $\PAut ( \CC_s ) = H_m$ in the remaining primitive cases treated here. The hull dimension is bounded by the genus on every regular stratum, apart from the type $(3,3,3)$. Explicit examples show that length, dimension, and hull dimension do not determine monomial equivalence, and that the hull profile together with the regular automorphism subgroup does not determine the curve up to isomorphism. As an application, we determine the corresponding hull-based invariants and punctured signatures used in support-splitting methods.
