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Generalized Hamming Weights of AJ-Gorenstein One-Point Codes
Eliseo Sarmiento-Rosales, José Alberto Guzmán-Vega, Juan Carlos Jiménez-Cervantes
TL;DR
Complete generalized-weight hierarchies are difficult to determine for one-point algebraic-geometric code flags because exact higher weights depend on rational-point geometry. The paper organizes these weights in a zero diagram and uses twisted and Wei duality plus geometric constructions to determine corresponding regions of the full flag. More than half of all generalized-weight positions are determined uniformly for AJ–Gorenstein triples with n > 2g, while the smallest Suzuki curve achieves a large exact coverage.
Problem
Exact higher-weight hierarchies are difficult to determine across one-point algebraic-geometric code flags because they depend on rational-point geometry.
Method
The paper organizes common zero sets in a zero diagram and combines twisted and Wei duality with divisor, gonality, support-packing, multiplication, and rank-propagation mechanisms.
Results
More than half of all generalized-weight positions in every AJ–Gorenstein complete flag with n > 2g are determined exactly.
Takeaways & Limitations
Each zero row simultaneously determines generalized weights for short codes and missing weights for reflected long-code partners.
Takeaways & Limitations
Gonality-based exactness depends on knowing the rational gonality sequence, which is curve-specific.
Abstract
from arXiv · showhide
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 $78.30\%$.
1 Introduction
The paper organizes generalized Hamming weights along an AJ–Gorenstein one-point-code flag through a zero diagram, whose rows connect short-code weights with reflected long-code weights. Geometric and duality-based mechanisms yield exact regions and uniform coverage exceeding half the complete flag.
- Motivation: The paper addresses the difficulty of determining complete weight hierarchies for many codes in one-point algebraic-geometric flags.Riemann–Roch provides robust bounds, but exact higher weights depend on rational-point geometry.
- Zero diagram: The zero diagram records, for each nongap-indexed code, the largest common zero sets of subcodes of prescribed dimensions.Its entries Mj,s are maximal numbers of evaluation points on which an s-dimensional subcode vanishes simultaneously.
- Duality correspondence: Each zero row determines both a lower block of short-code generalized Hamming weights and a reflected upper block of missing weights through twisted and Wei duality.The lower values are n − Mj,s, while the reflected missing weights are Mj,s + 1.
- Duality correspondence: The injective zero diagram is extended to the full range by reflection, reducing the complete-flag GHW problem to the injective region.In the postcanonical range, upper blocks can fold back into rows of the injective zero diagram.
- Geometric mechanisms: The postcanonical profile combines divisor reciprocity, gonality, full-support packings, multiplication, and rank propagation to produce exact zero-diagram regions.These mechanisms yield an exact tail, a full/no-full boundary, residual exactness, and full-level staircases.
- Coverage: More than half of all generalized-weight positions in every AJ–Gorenstein complete flag with n > 2g are determined exactly.The result is presented as a uniform coverage theorem, with the smallest Suzuki curve treated as a detailed example.
2 Preliminaries
This section introduces linear-code supports, generalized Hamming weights, duality, one-point evaluation codes, and the AJ–Gorenstein hypotheses governing the full zero-diagram range.
- Generalized Hamming weights measure the smallest support of a subcode at each dimension and form the code’s weight hierarchy.
- Wei duality identifies a code’s weights with the complement of reflected dual-code weights, supplying the missing positions in the hierarchy.
- One-point codes evaluate Riemann–Roch spaces L(hQ) on X, with the evaluation map injective for h<n.
- AJ–Gorenstein means DX ∼ nQ and symmetric H(Q), yielding N=n+2g−2 and a full zero-diagram interval 0≤h≤N.
3 The Coweight Calculus
The coweight calculus reformulates generalized Hamming weights through vanishing coordinates and establishes duality identities that organize coweight profiles.
- Coweights equal code length minus generalized Hamming weights, so coweight calculations reformulate Wei duality for arbitrary codes.
- Coweights can be characterized by the largest coordinate size supporting an s-dimensional subcode that vanishes there.
- Coweight profiles are strictly decreasing, mirroring the strictly increasing generalized-weight hierarchy.
- The kernel identity compares vanishing dimensions on a coordinate set with those of the scaled dual on its complement.
- For a scaled dual code, the coweight profile is the complement of reflected coweights from the original code.
4 The Injective Zero Diagram (λj < n)
For λj<n, the injective zero diagram records maximum vanishing numbers that equal coweights and determines complete weight hierarchies; geometric bounds and Cartesian embeddings fill its entries.
- Coweights and zero rows: For λj<n, row j records the generalized coweights of Cj= CX(λjQ), with dimension j+1 and vanishing-set interpretation.
- Coweights and zero rows: Twisted duality extends the injective diagram to λj≤N, determining long-code entries by reflection without new geometric input.
- Coweights and zero rows: The injective zero diagram determines the complete weight hierarchy of every code in the one-point flag through ds(Cj)=n−Mj,s.
- Cartesian profiles: Under a surjective constant-fiber Cartesian coordinate map, row t is completely determined by affine subspaces with qt−s solutions.
- Gonality bounds: The gonality sequence gives the bound Mj,s≤λj−γs−1(C), whose exact usefulness depends on curve-specific rational gonality data.
- Coweights and zero rows: Full-support and divisor constructions provide additional zero-diagram entries, including bounds and closure under disjoint unions and complements.
5 The Full Zero Diagram
The full zero diagram extends the injective diagram to all pole orders by twisted duality. Each zero row simultaneously determines generalized weights in a lower block and missing weights in its reflected upper block.
- Full zero diagram: The full zero diagram determines generalized Hamming weights for every one-point code CX(hQ) with 0 ≤ h ≤ N, including h ≥ n.For h ≥ n, evaluation is no longer injective, but the reflected construction still determines the GHW diagram.
- Twisted duality: Twisted duality identifies CX((N − h)Q) with a v-scaled dual of CX(hQ), with equality following from complementary dimensions.A differential with divisor NQ − DX supplies the nonzero coordinate multipliers v.
- Upper–lower block dictionary: A single zero row Mj determines both blocks: lower-block weights are n − Mj,s, while upper-block missing weights are Mj,s + 1.The upper block has dimension n − j − 1 and omits exactly the reflected values Mj,s + 1.
- Zero profiles: The zero profile satisfies mλjQ(e) = #{s ∈ {1, . . . , j + 1} : Mj,s ≥ e}, linking maximum vanishing numbers to each zero row.This follows by identifying the Riemann–Roch space with the code Cj and applying the coweight-profile lemma.
6 The Postcanonical Region and the Coverage Theorem
The postcanonical region yields exact coweights and generalized weights through reciprocity, gonality, and full-support conditions. Counting these exact cells gives a uniform coverage bound exceeding one half for the complete flag.
- Fold-back phenomenon: Postcanonical upper blocks fold back to the complementary row j∨ = n − j − 2 of the injective zero diagram.Because N − λj < n once λj > 2g − 2, the reflected block remains within the injective range.
- Postcanonical exactness: For 2g ≤ λj < n, the postcanonical zero row satisfies Mj,j+1−ρ = ρ for 0 ≤ ρ ≤ min{j, max{j − g, γ(C) − 2}}.The corresponding generalized weights are ds(Cj) = n − j − 1 + s for g + 1 ≤ s ≤ j + 1.
- Canonical boundary: The canonical boundary has an exact alternative: Mj,g equals a when the relevant support is full and a − 1 otherwise.Here a denotes the postcanonical parameter associated with λj.
- Staircases: Full-support divisors and gonality hypotheses extend exactness to staircase entries satisfying Mj,g−w = a + w.This gives dg−w(Cj) = n − λj + 2g − 2 − w under the stated conditions.
7 Specialization to Castle Curves
Castle curves supply AJ–Gorenstein structure together with uniform coordinate fibers, full-support packings, and gonality estimates. These mechanisms provide additional exact zero-diagram entries and sharpen coverage statements.
- Castle structure: Every Castle triple is AJ–Gorenstein, and its evaluation length is n = qλ1.The Castle function establishes the divisor relation required for the Abel–Jacobi condition, while semigroup symmetry gives the canonical condition.
- Cartesian rows: A surjective coordinate map with uniform fibers propagates exact first-column entries through all rows up to the Cartesian grade.Uniform fibers are essential for the displayed intersection counts; low-degree relations alone do not suffice.
- Universal fiber packing: The Castle function partitions X into disjoint fibers, producing a full packing and placing every multiple tλ1, 1 ≤ t ≤ q − 1, in ΦX.These packings yield exact first-column entries beyond the Cartesian grade.
- Gonality: For Castle curves, γ(C) = λ1 whenever λ1 ≤ q + 1.The upper bound comes from the Castle function, while rational-point counting supplies the matching lower bound.
- Postcanonical Castle rows: Rows with w0(aj) ≤ min{g − 1, γ(C) − 2} contain an exact staircase of length min{g − 1, γ(C) − 2} − w0(aj) + 1.The universal fiber packing supplies the required support value for applying the postcanonical theorem.
- Coverage specialization: For Castle curves with qλ1 > 2g, the general coverage bound specializes to q, λ1, and g, exceeding one half when qλ1 > 5g.The same asymptotic mechanism applies along families whose genus grows more slowly than qλ1.
8 Conclusion and Open Problems
The paper presents the zero diagram as a unified representation of generalized coweights and uses duality and geometric constructions to determine exact regions of the complete GHW flag. It also identifies unresolved coverage, geometric, and semigroup-determination questions.
- Conclusions: Zero-diagram rows encode maximal common zero sets and determine weights of short codes alongside missing weights of reflected long-code partners.Divisor reciprocity, gonality, full-support packings, multiplication, and rank propagation generate exact regions from finite geometric data.
- Main consequence: For every AJ–Gorenstein triple with n > 2g, more than half of all generalized-weight positions in the complete flag are determined exactly.The result concerns the GHW diagram, not the auxiliary injective zero diagram.
- Open problems: The paper leaves open how tight the coverage bounds are and what governs the cells that remain undetermined.Candidate explanations include gonality sequences, special divisors, and the full-support set ΦX.
- Semigroup dependence: It is unresolved whether the Weierstrass semigroup alone determines the GHW diagram for triples sharing n, g, and H(Q).The unresolved cells of the Suzuki example are proposed as a test case for this question.
A.1 The GHW diagram of the smallest Suzuki curve
The smallest Suzuki GHW diagram contains 2,912 generalized-weight positions across 91 one-point codes, and its coverage map marks the regions separated by h = 2g and h = n.
- Smallest Suzuki curve: 2,912 generalized-weight positions occur across the 91 codes C_X(hQ), for 0 ≤ h ≤ 90.The first complete code is C_X(91Q) = F_64.
- Coverage map: The coverage map uses dashed lines at h = 2g and h = n to partition the smallest Suzuki GHW diagram.Colors correspond to classes whose supplying theorems or corollaries are listed in Table 1.
A.2 Theorem-by-theorem coverage
Table 1 assigns disjoint cell counts by descending priority among applicable results, while the first result row is further decomposed into three named contributions.
- Assignment rule: Table 1 reports disjoint contributions because each cell is assigned to the first applicable class in descending order.The table's New column counts cells not already assigned to an earlier class.
- First result row: The first result row decomposes into an adaptive postcanonical tail, canonical-boundary gap cases, and an additional upper-tail cell.These components are identified respectively as the adaptive postcanonical tail, boundary gaps, and a degree-determined upper tail.
- First result row: 21 cells come from the adaptive tail, 7 from boundary gaps, and 1 from the upper tail.Together, these are the last three terms in the first result-row decomposition.