Unique Decoding of Reed-Solomon and Related Codes for Semi-Adversarial Errors

Joshua Brakensiek; Manik Dhar; Yeyuan Chen; Zihan Zhang

REVIEW 2 major objections 2 minor 1 cited by

Unique Decoding of Reed-Solomon and Related Codes for Semi-Adversarial Errors

T0 review · 2 major / 2 minor · reviewed 2026-05-22 · grok-4.3

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

Pith's one-line read Near-linear time unique decoding works for Reed-Solomon codes under semi-adversarial errors

desk verdict The paper introduces a semi-adversarial error model and gives near-linear unique decoders for IRS, FRS, and multiplicity codes that claim to match information-theoretic bounds, with monomial tracking as the main new tool. read the letter →

arxiv 2504.10399 v3 pith:JRZER62Q submitted 2025-04-14 cs.IT cs.DSmath.IT

Joshua Brakensiek , Yeyuan Chen , Manik Dhar , Zihan Zhang This is my paper

classification cs.ITcs.DSmath.IT

keywords Reed-Solomoncodesuniquedecodingsemi-adversarialerrorsinterleavedfoldedmultiplicitynear-lineartime

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper defines semi-adversarial errors as a setting where an adversary corrupts some code symbols while the remaining errors are uniformly random. It constructs near-linear time unique decoders for interleaved Reed-Solomon codes by refining an existing random-error algorithm through a monomial-tracking analysis. The same framework yields fast algorithms for folded Reed-Solomon and univariate multiplicity codes by introducing interpolations that replace root-finding steps with direct polynomial division. These decoders achieve the information-theoretic optimum across most mixtures of the two error types.

What carries the argument

Monomial-tracking technique that bounds decoder success when some symbols are chosen adversarially and the rest are random, together with new interpolations that enable polynomial long division in place of root finding.

What would settle it

An explicit semi-adversarial error pattern below the information-theoretic threshold on which the decoder fails to recover the correct message would disprove the claimed performance and optimality.

Watch

Extended reading notes

Core claim

For interleaved Reed-Solomon codes an improved version of the Bleichenbacher-Kiayias-Yung decoder, equipped with a novel monomial-tracking technique, performs unique decoding in near-linear time for most semi-adversarial error patterns. Novel interpolations extend the approach to folded Reed-Solomon and univariate multiplicity codes, replacing the root-finding step of prior algorithms with straightforward polynomial long division while preserving the same near-linear runtime and matching the information-theoretic optimum.

Load-bearing premise

The monomial-tracking analysis and new interpolations accurately predict decoder success on error patterns that mix adversarial and random corruptions.

Editorial extensions

If this is right

Interleaved Reed-Solomon codes admit near-linear unique decoding against most semi-adversarial mixtures via monomial tracking.
Folded Reed-Solomon codes admit near-linear unique decoding via interpolations that reduce to polynomial division.
Univariate multiplicity codes admit near-linear unique decoding under the same mixed error model.
All three families achieve the information-theoretic optimum for the semi-adversarial setting.
The new techniques simplify traditional decoding pipelines while retaining optimal error correction.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The monomial-tracking method may apply to other algebraic codes whose decoders track coefficient growth.
Systems facing hybrid random-plus-adversarial noise could adopt these decoders for efficiency without sacrificing optimality.
Parameter regimes where the random fraction dominates may admit even simpler implementations.

Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Desk Editor's Note

read the letter

The main point is that they define semi-adversarial errors as a mix of random and adversarial corruptions, then build near-linear time unique decoders for interleaved Reed-Solomon, folded Reed-Solomon, and univariate multiplicity codes that reach the optimal radius for most mixtures. The monomial-tracking technique lets them analyze the BKY algorithm in this setting for IRS, and they adapt the same idea to FRS and multiplicity codes by swapping root-finding for simple polynomial long division. That change is a concrete simplification over Guruswami-Wang style methods. The work sits squarely on prior BKY and Guruswami-Wang results, so the foundations are clear and the extensions are the actual addition. If the analysis is tight, the result is useful for any setting where errors split between noise and targeted jamming. The soft spot is the handling of possible correlations. The monomial-tracking bounds rely on probabilistic counting that assumes the random and adversarial symbols behave independently enough for the degree distributions to work out. If the adversary can pick its symbols after seeing the random ones or the codeword, those counts could shift and the claimed optimality for “most mixtures” would only hold under a non-adaptive adversary. The paper needs to state this assumption explicitly and show the bounds survive it. Runtime claims for the full parameter ranges also want direct verification. This is for coding theorists who care about efficient algebraic decoders and hybrid error models. A reader working on list or unique decoding algorithms would get concrete constructions and a new analysis trick to look at. It deserves peer review because the claims are specific enough to check and the techniques are grounded in existing frameworks.

Referee Report

2 major / 2 minor

Summary. The paper introduces the semi-adversarial error model, where an arbitrary subset of codeword symbols may be corrupted by an adversary while the remainder are replaced by uniformly random symbols. It presents near-linear-time unique decoding algorithms for interleaved Reed-Solomon (IRS) codes by refining the Bleichenbacher-Kiayias-Yung (BKY) algorithm with a novel monomial-tracking technique, and extends the approach to folded Reed-Solomon (FRS) and univariate multiplicity codes via new interpolation procedures that replace the root-finding step of prior algorithms (e.g., Guruswami-Wang) with polynomial long division. The analyses are claimed to achieve the information-theoretic optimum for most mixtures of random and adversarial errors.

Significance. If the monomial-tracking arguments are shown to be rigorous, the work supplies the first efficient unique decoders that interpolate between fully random and fully adversarial regimes for these code families, with a concrete simplification (long division in place of root-finding) that may be of independent interest. The explicit matching to information-theoretic bounds and the constructive use of the BKY and Guruswami-Wang frameworks are strengths.

major comments (2)

[IRS monomial-tracking analysis] IRS decoder analysis (monomial-tracking section): the probabilistic counting arguments that deliver the information-theoretic optimum must explicitly address whether the adversary may select its symbols after observing the random symbols and the codeword; if adaptive choice is permitted, the claimed independence between random and adversarial positions used to bound the failure probability requires a separate justification or counter-example, as correlations could invalidate the degree-distribution tracking.
[FRS and multiplicity extensions] FRS and multiplicity-code extensions: the claim that the new interpolations yield near-linear runtime while preserving optimality rests on 'more robust monomial-tracking arguments' than for IRS; however, the manuscript provides no explicit comparison of the resulting error-probability bounds to those of Guruswami-Wang, nor a parameter-range verification that the long-division step does not increase the failure probability beyond the stated optimum.

minor comments (2)

[Notation] Notation for the semi-adversarial parameter (fraction of adversarial symbols) should be introduced once and used consistently; currently it appears under multiple symbols in the abstract and main text.
[Results summary] The manuscript would benefit from a short table summarizing the achieved rate-error trade-offs for each code family under the semi-adversarial model.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and insightful comments on our manuscript. We address the two major comments point by point below, clarifying the error model and outlining planned revisions to strengthen the presentation.

read point-by-point responses

Referee: [IRS monomial-tracking analysis] IRS decoder analysis (monomial-tracking section): the probabilistic counting arguments that deliver the information-theoretic optimum must explicitly address whether the adversary may select its symbols after observing the random symbols and the codeword; if adaptive choice is permitted, the claimed independence between random and adversarial positions used to bound the failure probability requires a separate justification or counter-example, as correlations could invalidate the degree-distribution tracking.

Authors: We thank the referee for highlighting this subtlety. The semi-adversarial error model in the paper defines the adversary as selecting an arbitrary subset of positions to corrupt arbitrarily, with the remaining positions replaced by uniformly random symbols chosen independently of both the codeword and the adversary's choices. This is a non-adaptive model in which the random symbols are generated without the adversary observing them in advance. Consequently, the independence between random and adversarial positions holds, and the monomial-tracking arguments for bounding the failure probability remain valid. We will add an explicit clarification of the non-adaptive nature of the adversary in the revised manuscript. revision: yes
Referee: [FRS and multiplicity extensions] FRS and multiplicity-code extensions: the claim that the new interpolations yield near-linear runtime while preserving optimality rests on 'more robust monomial-tracking arguments' than for IRS; however, the manuscript provides no explicit comparison of the resulting error-probability bounds to those of Guruswami-Wang, nor a parameter-range verification that the long-division step does not increase the failure probability beyond the stated optimum.

Authors: We appreciate the referee's suggestion for greater explicitness. Although the monomial-tracking arguments for FRS and multiplicity codes are designed to be more robust to accommodate the folded and multiplicity structures, the manuscript does not provide a direct side-by-side comparison of the resulting error-probability bounds against the Guruswami-Wang algorithm, nor a parameter-range check confirming that the long-division step preserves the stated optimality. In the revision we will add such a comparison and verification that the long-division replacement does not increase the failure probability beyond the information-theoretic optimum in the relevant regimes. revision: yes

Circularity Check

0 steps flagged · score 0.0 of 10

Derivation chain is self-contained with novel analysis techniques

full rationale

The paper introduces the semi-adversarial error model and develops decoding algorithms for IRS, FRS, and multiplicity codes via a novel monomial-tracking technique and new interpolations that replace root-finding steps. It explicitly builds on the external BKY algorithm and Guruswami-Wang framework but supplies independent analysis and extensions without any reduction of claims to self-fitted parameters, self-definitional loops, or load-bearing self-citations. No equations or steps in the provided text reduce the central results to their own inputs by construction.

Assumptions & free parameters 0 free parameters · 1 assumptions · 1 invented entities

Based on the abstract alone, the central claim rests on standard properties of Reed-Solomon codes plus the correctness of the newly introduced monomial-tracking analysis for the mixed error model; no explicit free parameters or invented physical entities are mentioned.

assumptions (1)

standard math Standard algebraic properties of Reed-Solomon, interleaved Reed-Solomon, folded Reed-Solomon, and multiplicity codes hold.
The algorithms and analysis presuppose these well-known code properties from prior literature.

invented entities (1)

semi-adversarial error model
purpose: To interpolate between fully random and fully adversarial symbol errors
New error model introduced to motivate the decoding algorithms.

how reviews work

0 comments

Cite this review

Pith. "Pith review of Unique Decoding of Reed-Solomon and Related Codes for Semi-Adversarial Errors." pith.science (2026). https://pith.science/paper/JRZER62Q

@misc{pith2026250410399,
  author       = {Pith},
  title        = {Pith review of: Unique Decoding of Reed-Solomon and Related Codes for Semi-Adversarial Errors},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/JRZER62Q}},
  note         = {Machine review of arXiv:2504.10399}
}

read the original abstract

Motivated by recent developments in coding theory, particular in list-decoding, we introduce a new error model which we call semi-adversarial errors. This error model bridges between fully random errors and fully adversarial errors by allowing some symbols of a message to be corrupted by an adversary while others are replaced with uniformly random symbols. As our main quest, we seek to understand optimal efficient unique decoding algorithms in the semi-adversarial model. For interleaved Reed--Solomon (IRS), folded Reed--Solomon (FRS) and univariate multiplicity codes, we design decoding algorithms running in near-linear time for most mixtures of random and adversarial errors. Our analysis matches the information-theoretic optimum for semi-adversarial errors. Our algorithm for interleaved Reed--Solomon codes is an improved implementation of the decoding algorithm by Bleichenbacher--Kiayias--Yung (BKY) for fully random errors. We use a novel monomial-tracking technique to analyze its performance in this new semi-adversarial errors. Inspired by the BKY algorithm, we use novel interpolations to extend our approach to the settings of folded Reed--Solomon and multiplicity codes, resulting in fast algorithms for unique decoding against semi-adversarial errors. Our new decoders for FRS and multiplicity codes replace the sophisticated root-finding step in traditional algorithms, such as the Guruswami--Wang algorithm, with a straightforward polynomial long division. Analysis of these algorithms requires more robust monomial-tracking arguments than IRS codes.

Discussion (0). Sign in to comment.

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/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

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

We use a novel monomial-tracking technique to analyze its performance in this new semi-adversarial errors... Bounding Matrix Rank: Proof of Theorem 3.6
IndisputableMonolith/Foundation/ArithmeticFromLogic.lean embed_injective unclear

?

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

the block matrix B ... has full column rank

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.

Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

Simultaneous Rational Function Codes: Improved Analysis Beyond Half the Minimum Distance with Multiplicities and Poles
cs.IT 2025-08 unverdicted novelty 5.0 of 10

The abstract promises improved failure-probability analysis for rational-function decoding, but the body text is a different paper.

Pith tools

Reviewed May 22, 2026 · model on record in the stance chip above.