Generalized Hamming Weights of AJ-Gorenstein One-Point Codes
Information Theory
Summary
The authors study special kinds of error-correcting codes derived from a type of algebraic curve called an AJ-Gorenstein curve. They organize the codes' properties in a structured way to better understand important parameters called generalized Hamming weights, which measure the error-detection capability of the codes. They prove that for curves of genus g and sufficiently large evaluation length n, more than half of the key weight positions can be exactly determined in a uniform manner. Applying their results to a specific example, the Suzuki curve, they show that their methods determine over 78% of these positions exactly. This work helps clarify how much of the code's structure can be precisely understood using dualities and diagrammatic tools.
Authors
Eliseo Sarmiento-Rosales, José Alberto Guzmán-Vega, Juan Carlos Jiménez-Cervantes
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 $\operatorname{Cov}_{\mathrm{full}} \ge \frac{n(n-1)+4g}{n(n+2g-1)}>\frac12$. 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 $78.30\%$.