Generalized Hamming Weights of AJ-Gorenstein One-Point Codes
Abstract
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 Covfull ≥ (n(n−1)+4g)/(n(n+2g−1))>1/2. 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%.
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BibTeX
@misc{sarmientorosales2026generalized,
title = {{Generalized Hamming Weights of AJ-Gorenstein One-Point Codes}},
author = {Eliseo Sarmiento-Rosales and José Alberto Guzmán-Vega and Juan Carlos Jiménez-Cervantes},
howpublished = {arXiv},
year = {2026},
month = aug,
doi = {10.48550/arxiv.2608.29508},
eprint = {2608.29508},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.29508},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Generalized Hamming Weights of AJ-Gorenstein One-Point Codes
AU - Eliseo Sarmiento-Rosales
AU - José Alberto Guzmán-Vega
AU - Juan Carlos Jiménez-Cervantes
PY - 2026
DA - 2026/08/30
PB - arXiv
DO - 10.48550/arxiv.2608.29508
UR - https://arxiv.org/abs/2608.29508
AB - 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 Cov_full ≥ (n(n−1)+4g)/(n(n+2g−1))>1/2. 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%.
ER -