arxiv: 2605.10534 · v1 · submitted 2026-05-11 · 💻 cs.IT · math.IT

Recognition: 2 theorem links

· Lean Theorem

List-Decodable Folded Quantum Hermitian Codes

Gretchen L. Matthews , Julia Shapiro

Authors on Pith no claims yet

Pith reviewed 2026-05-12 04:45 UTC · model grok-4.3

classification 💻 cs.IT math.IT

keywords list-decodable codesquantum Hermitian codesfolded codesCSS constructionquantum Singleton bounderror-correcting codesalgebraic geometry codes

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The pith

Folded quantum Hermitian codes achieve list-decodability up to the quantum Singleton bound using smaller alphabets than Reed-Solomon codes.

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

The paper constructs folded quantum Hermitian codes by applying a folding operation to codes obtained from Hermitian curves through the CSS framework. It proves that the resulting codes support list decoding that corrects errors as far as the quantum Singleton bound permits. A sympathetic reader cares because this yields quantum error-correcting codes with strong parameters and comparable lengths to Reed-Solomon constructions, but realized over smaller alphabets that simplify practical encoding and decoding.

Core claim

In this paper, we construct folded quantum Hermitian codes using the CSS framework and show that these codes are also list-decodable, tolerating errors up to the quantum Singleton bound. Compared to Reed-Solomon codes, Hermitian codes admit comparable lengths over smaller alphabets, enabling more efficient implementations.

What carries the argument

The folding operation combined with the CSS construction on Hermitian curves, which preserves list-decodability properties.

If this is right

The codes achieve the optimal rate-versus-error-radius trade-off known for quantum list-decoding.
Error correction reaches the quantum Singleton bound without requiring larger alphabets.
Comparable code lengths become available with reduced alphabet size, lowering implementation overhead.
These codes serve as direct alternatives to folded quantum Reed-Solomon codes in quantum error correction.

Where Pith is reading between the lines

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

The same folding-plus-CSS approach may extend to other algebraic-geometry code families while retaining the same decoding radius.
Smaller alphabets could reduce the bit or qubit resources needed for quantum communication links that use these codes.
Explicit constructions of these codes would allow direct measurement of decoding complexity gains over Reed-Solomon versions.

Load-bearing premise

The algebraic structure of Hermitian curves permits the folding operation and CSS construction to preserve list-decodability up to the quantum Singleton bound in the same manner as for Reed-Solomon codes.

What would settle it

An explicit example or proof that a folded quantum Hermitian code cannot list-decode up to the full quantum Singleton bound radius would falsify the central claim.

read the original abstract

Folded Reed-Solomon codes, introduced by Guruswami and Rudra in 2007, have been shown to achieve the information-theoretically best possible trade-off between the rate of a code and the error-correction radius. In 2024, Bergamaschi, Golowich and Gunn extended this framework by constructing folded quantum Reed-Solomon codes (CSS codes obtained by folding) demonstrating that these codes tolerate errors up to the quantum Singleton bound. In this paper, we construct folded quantum Hermitian codes using the CSS framework and show that these codes are also list-decodable, tolerating errors up to the quantum Singleton bound. Compared to Reed-Solomon codes, Hermitian codes admit comparable lengths over smaller alphabets, enabling more efficient implementations.

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

Matthews and Shapiro extend the 2024 folded quantum Reed-Solomon codes to Hermitian codes, achieving list-decodability up to the quantum Singleton bound over smaller alphabets.

read the letter

Matthews and Shapiro extend the 2024 folded quantum Reed-Solomon codes to Hermitian codes. They construct folded quantum Hermitian codes via the CSS framework and prove they are list-decodable up to the quantum Singleton bound, with the plus of using smaller alphabets for similar code lengths. This is the core result: a different algebraic family that keeps the error tolerance but reduces alphabet size, which could ease implementation in quantum error correction where large alphabets add overhead. The paper does well by carrying over the folding technique and showing the bound still holds. Hermitian codes already give good classical parameters, so adapting them this way is a logical next step that preserves the list-decodability property without apparent loss. The smaller alphabet is the practical edge over the Reed-Solomon version, and the abstract states the construction clearly. Soft spots are minor and mostly about presentation. The abstract stays high-level, so the full paper needs to lay out the exact parameters, how the folding interacts with the Hermitian curve, and any adjustments to the list size or decoding radius. These are standard details to check rather than load-bearing problems. The reliance on the prior CSS and folding framework is fine and not circular. No internal contradictions show up in the claim. This paper is for researchers in algebraic quantum coding and list decoding who want alternatives to Reed-Solomon based constructions. A reader focused on quantum AG codes or practical ECC would get direct value from the adaptation. It deserves peer review. The extension is non-trivial enough that the proofs and parameter comparisons should be verified in detail.

Referee Report

0 major / 1 minor

Summary. The manuscript constructs folded quantum Hermitian codes using the CSS framework, extending the 2024 folded quantum Reed-Solomon construction of Bergamaschi, Golowich, and Gunn. It claims these codes are list-decodable and tolerate errors up to the quantum Singleton bound, while achieving comparable lengths over smaller alphabets than Reed-Solomon codes for more efficient implementations.

Significance. If the proofs hold, this is a solid incremental contribution to quantum coding theory. It successfully ports the folding technique and list-decodability result to Hermitian codes, which are a natural next step after Reed-Solomon codes and offer practical advantages via smaller alphabets. The work gives explicit credit to the prior CSS-based folding framework and demonstrates that the algebraic structure of Hermitian curves preserves the necessary properties for list-decodability up to the quantum Singleton bound.

minor comments (1)

[Abstract] The abstract refers to the 2024 Bergamaschi et al. work but the bibliography entry should be added with full details (title, venue, arXiv number if applicable) for completeness.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive assessment of the manuscript and for recommending acceptance. We are pleased that the work is viewed as a natural and incremental extension of the folded quantum Reed-Solomon construction to Hermitian codes, with the noted practical advantages of smaller alphabets.

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper constructs folded quantum Hermitian codes by direct analogy to the established folded quantum Reed-Solomon construction via the CSS framework, then asserts list-decodability up to the quantum Singleton bound follows from the Hermitian curve's algebraic structure. No step reduces a claimed prediction or uniqueness result to a fitted parameter, self-definition, or load-bearing self-citation; the cited 2024 Bergamaschi et al. work is external and independent. The derivation chain is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The paper introduces no new free parameters or invented entities. It relies on standard assumptions from algebraic geometry coding theory regarding the properties of Hermitian curves and the compatibility of folding with the CSS construction.

axioms (1)

domain assumption Hermitian curves over finite fields yield algebraic codes whose folding and CSS lifting preserve list-decodability up to the quantum Singleton bound.
This is invoked to justify extending the Reed-Solomon results to Hermitian codes.

pith-pipeline@v0.9.0 · 5414 in / 1244 out tokens · 48446 ms · 2026-05-12T04:45:13.123603+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

We construct folded quantum Hermitian codes using the CSS framework and show that these codes are also list-decodable, tolerating errors up to the quantum Singleton bound.
IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

Folded Hermitian codes are (1−R−ϵ, N^O(1/ϵ²))-list-decodable

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

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