arxiv: 2605.01348 · v1 · submitted 2026-05-02 · 🧮 math.AG · cs.IT· math.IT

Weierstrass semigroups and the order bound

Alix Barraud , Ya\u{g}mur \c{C}ak{\i}ro\u{g}lu , Bianca Gouthier , Gretchen L. Matthews , Lara Vicino This is my paper

Pith reviewed 2026-05-09 18:40 UTC · model grok-4.3

classification 🧮 math.AG cs.ITmath.IT

keywords Weierstrass semigroupsStöhr-Voloch theoryFeng-Rao boundorder boundalgebraic geometry codesone-point codesminimum distance

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The pith

Weierstrass semigroups determine the Feng-Rao order bound for one-point algebraic geometry codes.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This survey provides an accessible introduction to Weierstrass semigroups within the theory developed by Stöhr and Voloch. It shows that these semigroups supply the data needed to compute the order bound, also known as the Feng-Rao bound, on the dual minimum distance of one-point algebraic geometry codes from a curve. A reader would care because the bound gives a concrete way to estimate code performance from the curve's local function orders rather than from exhaustive codeword searches. The presentation assumes basic familiarity with curves over finite fields and linear codes.

Core claim

In the Stöhr-Voloch framework the Weierstrass semigroup at a rational point on the curve consists of the possible pole orders of rational functions that are regular everywhere except at that point. This semigroup directly determines the order bound for the dual of any one-point algebraic geometry code constructed from the same point, yielding an explicit lower estimate on the minimum distance.

What carries the argument

The Weierstrass semigroup at a point on the curve, the numerical semigroup of non-negative integers realized as pole orders of functions regular away from the point.

If this is right

Once the Weierstrass semigroup is known, the order bound can be calculated without enumerating all codewords.
The same semigroup data applies to every one-point code constructed from the given point on the curve.
Curves whose semigroups are explicitly known, such as the Hermitian curve, admit immediate computation of the order bound for their associated codes.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The survey makes it feasible to compare the order bound across different curves by first tabulating their semigroups at chosen points.
The same semigroup machinery may suggest how to tighten other distance bounds that also depend on pole-order information.

Load-bearing premise

The reader already has enough background in algebraic geometry and coding theory to follow an essential introduction to the topic.

What would settle it

A concrete one-point algebraic geometry code whose dual minimum distance cannot be bounded using only the Weierstrass semigroup at the evaluation point would show the claimed reliance does not hold.

read the original abstract

The aim of this survey is to provide the reader with an essential and accessible introduction to the theory of Weierstrass semigroups, in the context of the theory developed by K.-O. St\"ohr and J.F. Voloch. Furthermore, we discuss an application of St\"ohr-Voloch theory in coding theory, namely the Feng-Rao bound (also known as the order bound) for the dual minimum distance of one-point algebraic geometry codes from a curve, which relies on the knowledge of certain Weierstrass semigroups of the curve.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

This is a clear survey of Weierstrass semigroups in the Stöhr-Voloch setting and their link to the Feng-Rao bound, but it adds no new results.

read the letter

This paper is a survey that explains Weierstrass semigroups in the Stöhr-Voloch framework and their role in the Feng-Rao bound for one-point algebraic geometry codes. It introduces no new results. What it does well is organize the standard material into a coherent and readable introduction. The connection between the semigroup properties and the bound on the dual distance is laid out step by step, which can save time for readers who would otherwise piece it together from multiple sources. The soft spots are minor and expected for a survey. It does not claim originality, so its value rests entirely on the quality of the exposition. The abstract mentions an essential and accessible introduction, but the reader still needs prior knowledge of curves, divisors, and basic coding theory to follow along. There are no hidden assumptions or circular arguments since everything traces back to established literature. This paper is for readers who want a compact entry point into this specific intersection of algebraic geometry and coding theory. A graduate student or early-career researcher working on AG codes would get the most out of it. Experts in the area will already know the content. I would consider bringing it to a reading group if the group is exploring applications of curves to codes. I would not cite it for any new contribution. It deserves peer review in a venue that publishes surveys, as the work appears careful and the topic is well-defined.

Referee Report

0 major / 1 minor

Summary. The manuscript is an expository survey whose aim is to provide an essential and accessible introduction to Weierstrass semigroups in the Stöhr-Voloch framework, together with their role in the Feng-Rao (order) bound for the dual minimum distance of one-point algebraic geometry codes.

Significance. If the exposition is accurate, the survey assembles standard background material on Weierstrass semigroups and the order bound without advancing new theorems or quantitative claims; its potential value is therefore limited to serving as a clear reference for readers who already possess the requisite background in algebraic geometry and coding theory.

minor comments (1)

The abstract states the scope clearly, but the manuscript should verify that all cited definitions and examples from Stöhr-Voloch theory are presented without gaps so that the claimed accessibility is achieved.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their positive review and recommendation to accept the manuscript. The referee accurately characterizes the work as an expository survey on Weierstrass semigroups in the Stöhr-Voloch framework and its application to the Feng-Rao (order) bound. We address the significance assessment below.

read point-by-point responses

Referee: If the exposition is accurate, the survey assembles standard background material on Weierstrass semigroups and the order bound without advancing new theorems or quantitative claims; its potential value is therefore limited to serving as a clear reference for readers who already possess the requisite background in algebraic geometry and coding theory.

Authors: We agree that the manuscript is an expository survey compiling standard results from the literature without introducing new theorems or quantitative improvements. Its intended contribution is to offer a focused and accessible presentation of Weierstrass semigroups specifically within the Stöhr-Voloch approach, together with their direct use in deriving the order bound for one-point algebraic geometry codes. We believe this framing provides a useful reference for readers who may be familiar with algebraic geometry codes but less so with the Stöhr-Voloch perspective on semigroups. revision: no

Circularity Check

0 steps flagged

Expository survey with no internal derivations or predictions

full rationale

This manuscript is explicitly framed as a survey providing an accessible introduction to established Stöhr-Voloch theory on Weierstrass semigroups and recalling their role in the known Feng-Rao bound for one-point AG codes. No new theorems, derivations, quantitative predictions, or fitted parameters are advanced. All content assembles standard external background without any self-referential reduction of a claimed result to an input defined inside the paper. The reader's assessment of circularity score 0.0 is confirmed by the absence of any load-bearing steps that could be examined for circularity.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

The work is a survey of established mathematical theory and therefore inherits standard axioms from algebraic geometry (e.g., properties of Riemann-Roch spaces on curves) and coding theory without introducing or fitting new parameters or entities.

pith-pipeline@v0.9.0 · 5416 in / 1064 out tokens · 28384 ms · 2026-05-09T18:40:38.747202+00:00 · methodology

discussion (0)

Reference graph

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