AJ-Gorenstein一点码的广义汉明重量

中文摘要 AI 辅助

我们研究AJ-Gorenstein曲线上一点码旗标下的广义汉明重量，将这些重量组织为一个称为零图的分级数组，其元素是广义余重量：即指定维数的子码同时消失的评估点的最大数量。扭曲对偶和Wei对偶表明，零图的每一行既确定了短码低块的广义汉明重量，也确定了完整旗标GHW图中反射长码高块的缺失重量。我们的主要定量结果是一个统一覆盖定理：对于亏格为g且评估长度n>2g的AJ-Gorenstein曲线，完整旗标中精确确定的广义重量位置比例满足$\text{Cov}_{\text{full}} \ge \frac{n(n-1)+4g}{n(n+2g-1)}>\frac12$，即完整旗标中超过一半的广义重量位置被统一确定。对于最小的Suzuki曲线，通用机制和特定Castle机制共同确定了2912个位置中的2280个，精确覆盖率为78.30%。

英文摘要

We study generalized Hamming weights along the one-point code flag of an AJ-Gorenstein curve. We organize these weights in a graded array, the zero diagram, whose entries are generalized coweights: the largest numbers of evaluation points on which subcodes of prescribed dimensions vanish simultaneously. Twisted and Wei duality show that each row of the zero diagram determines both the generalized Hamming weights of a lower block of short codes and the missing weights of a reflected upper block of long codes in the GHW diagram of the complete flag. Our main quantitative result is a uniform coverage theorem. For an AJ-Gorenstein curve of genus $g$ and evaluation length $n>2g$, the proportion of generalized-weight positions determined exactly throughout the complete flag satisfies $\operatorname{Cov}_{\mathrm{full}} \ge \frac{n(n-1)+4g}{n(n+2g-1)}>\frac12$. Thus more than half of all generalized-weight positions in the complete flag are determined uniformly. For the smallest Suzuki curve, the general and Castle-specific mechanisms together determine $2{,}280$ of the $2{,}912$ positions, giving an exact coverage of $78.30\%$.

发表机构

• Escuela Superior de Física y Matemáticas, Instituto Politécnico Nacional（国立理工学院高等物理与数学学院）

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