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An Analysis of RPA Decoding of Reed-Muller Codes Over the BSC

T0 review · 2 major / 5 minor · reviewed 2026-08-11 · deepseek-v4-flash

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Pith's one-line read The RPA decoder provably drives Reed-Muller error probability to zero for orders up to roughly log m.

desk verdict First real analysis of RPA decoding error over the BSC, and the asymptotic claim is likely right, but the main concentration lemma is proved from an identity that only holds under the very conditioning it claims to avoid. read the letter →

arxiv 2412.08129 v2 pith:ESCCMJUA submitted 2024-12-11 cs.IT math.IT

V. Arvind Rameshwar , V. Lalitha This is my paper · ORCID
classification cs.ITmath.IT MSC 94B3594B05
keywords RPAdecoderReed-Mullercodesbinarysymmetricchannelerrorprobabilityboundprojection-aggregationtreefastHadamardtransformvanishingrecursivedecoding
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's claim is that the Recursive Projection-Aggregation (RPA) decoder, a practical algorithm previously supported mainly by simulations, can be proved to drive Reed-Muller error probability to zero on the binary symmetric channel (BSC) for code orders $r$ up to roughly $\log m$, where $N=2^m$ is the blocklength. Its main result is the explicit upper bound $P_{\mathrm{err}}(\mathrm{RM}(m,r)) \le 32 N^{r+1} \exp(-2^{-r-1} N \epsilon^2)$, valid for any $\epsilon < \eta(p) = \frac12(1-2p)^{2^{r-1}}$. A direct corollary is that every $r \le \log(c m)$ with $c < \log 2 / \log(1/(1-2p))$ has vanishing error as $m \to \infty$, and that fixed orders $r \ge 2$ always do. The proof bounds the two RPA subroutines\textemdash maximum-likelihood decoding of first-order RM projections and the majority-style aggregation step\textemdash and composes those bounds along the decoder's recursion tree, restricted to the event that every recursive call converges in one iteration.

What carries the argument

The projection-aggregation tree is the organizing device: each node is an RM code, each downward move replaces a node by its projections onto subspaces (one-dimensional in the main proof), and each upward move recombines decoded child estimates by the aggregation vote $\phi(\mathbf{x}) = \sum_i \mathbf{1}\{Y/B_i([\mathbf{x}+B_i]) \ne \hat{Y}/B_i([\mathbf{x}+B_i])\}$, flipping a received symbol when more than half the projections disagree. Two concentration estimates carry the argument: one shows that the Fast Hadamard Transform (ML) decoder on a first-order RM projection concentrates around the true Walsh character, and one shows that the aggregation statistic $\phi(\mathbf{x})$ concentrates around $p(1-Y_{\mathbf{x}})+(1-p)Y_{\mathbf{x}}$. Composing the resulting per-level failure probabilities along the tree, with the codeword-symmetry reduction to the all-zeros word, yields the bounds of Theorems III.1 and III.2.

What would settle it

Run the RPA decoder on a small RM code, for example $\mathrm{RM}(8,2)$ over BSC$(0.1)$, with both the all-zeros codeword and a random nonzero codeword, using enough trials to distinguish a factor-of-two difference in error rate; a systematic gap would refute the codeword-symmetry reduction on which the bounds rest. At a larger blocklength, compare the measured single-iteration failure rate against the theorem's exponential rate $2^{-r-1} N \epsilon^2$: if the decay is slower than linear in $N$, the central bound is wrong.

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Extended reading notes

Core claim

The authors establish that a single iteration of the RPA decoder at every level of its recursion is enough to give a provable error-decay guarantee. Working over the BSC and using one-dimensional projection subspaces, Theorem III.1 bounds $P_{\mathrm{err}}(\mathrm{RM}(m,r))$ by $32 N^{r+1} e^{-2^{-r-1} N \epsilon^2}$ for any $\epsilon<\eta(p)$, which yields the logarithmic-order corollary and the decoding-radius statement $N/2(1-\gamma_m-\delta)$ for fixed $r$. The same machinery, with Lipschitz concentration in place of Hoeffding, gives a parallel bound for $k$-dimensional subspaces (Theorem III.2) and shows the growth-rate guarantee for $r$ does not improve. The paper is explicit that the results cover only low-rate RM codes: the guaranteed orders have rate tending to zero, and whether RPA achieves positive rate or capacity under the BSC remains open.

Load-bearing premise

The load-bearing premise is that the error probability is independent of which RM codeword was sent, so bounding the all-zeros word bounds every codeword, and the proof establishes its concentration estimates only for that all-zeros input and only for the event that each recursive RPA call converges in a single iteration.

Editorial extensions

If this is right

  • Every fixed order $r\ge 2$ becomes provably decodable by RPA with vanishing error probability as the blocklength grows.
  • Orders $r \le \log(c m)$, for $c$ below $c(p)=\log 2/\log(1/(1-2p))$, also have vanishing error; these codes have vanishing rate, so the guarantee is for low-rate transmission.
  • For fixed $r$, the decoder corrects $N/2(1-\gamma_m-\delta)$ bit flips with high probability for large $m$, giving a concrete decoding-radius statement comparable to prior recursive decoders.
  • Using $k$-dimensional projection subspaces yields the same asymptotic growth of $r$ with $m$ as the one-dimensional case under this analysis, so higher-dimensional projections do not improve the proven order threshold.
  • Because the bounds charge any recursive call that fails to converge in one iteration as an error, the theorem's guarantees remain valid\textemdash and can only be conservative\textemdash for the full RPA decoder with more iterations.

Reading between the lines

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

  • The single-iteration restriction is the main bottleneck: if correlations across RPA iterations could be controlled, extending the recursion beyond one iteration is the natural route toward orders linear in $m$ and possibly positive rates, a direction the authors flag as future work.
  • The codeword-symmetry proposition is imported rather than re-proved, so a small-scale simulation comparing all-zeros and nonzero RM codewords under RPA would be the cheapest check of the proof's foundation.
  • The prefactor $N^{r+1}$ makes the bound's literal value unusably large at moderate blocklengths; the testable content is the exponential decay rate, and simulations should focus on the slope of $\log P_{\mathrm{err}}$ versus $N$ rather than on the constant.
  • Lemma IV.5 predicts a concrete two-iteration improvement for second-order codes\textemdash a factor-$N$ saving in the failure probability with the same exponent\textemdash which could be verified or contradicted by simulation before any higher-order extension is attempted.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 5 minor

Summary. The paper analyzes the recursive projection-aggregation (RPA) decoder of Ye and Abbe for Reed-Muller codes over the binary symmetric channel. The main result, Theorem III.1, is an explicit upper bound on the average error probability of RM(m,r) under single-iteration RPA decoding with one-dimensional projections: P_err <= 32 N^{r+1} exp(-2^{-r-1} N epsilon^2) for r >= 2 and 0 < epsilon < eta(p), where eta(p) is expressed in terms of the projected noise level. A corollary gives vanishing error probability for orders r up to roughly log m. The proof proceeds by bounding the ML/FHT error probability for first-order RM projections, analyzing the aggregation step on the event that all projections decode to zero, and then recursively composing these estimates up a projection-aggregation tree. A separate theorem extends the bound to k-dimensional projections.

Significance. If the main theorem is correct, the paper gives the first theoretical guarantee that the RPA decoder, a practical algorithm, achieves vanishing error probability on RM codes of growing order over the BSC. The explicit non-asymptotic bound and the clean separation of the FHT/ML base case, the aggregation step, and the recursive composition are strengths. The proof is self-contained apart from the standard concentration inequalities and the codeword-symmetry fact from [23]. No parameters are fitted; epsilon and delta are arbitrary in their stated ranges. However, the concentration argument for the aggregation step has a load-bearing gap (see major comments), so the manuscript in its current form does not fully establish the main claim.

major comments (2)
  1. [IV-B, Lemma IV.4 and Proposition IV.2] Lemma IV.4 is stated as an unconditional concentration bound for phi^(N)(x), but its proof uses identity (6), which is valid only on the event G that every FHT projection is decoded to the all-zero word. Off G the actual count phi(x) = sum_i 1{Y/B_i != hat-Y/B_i} can differ from the right-hand side of (6), and the proof gives no bound for that difference. Read as a conditional statement on G, the Hoeffding step would require the coordinates of Y to be i.i.d. after conditioning on G, which is not available. Proposition IV.2 then uses Lemma IV.4 through the law of total probability to get an unconditional closeness-of-indicators bound, and converts it to the conditional bound on G by subtracting Pr(G^c). The missing ingredient is an unconditional concentration statement for the actual phi(x), or a separate bound on the probability that phi(x) differs from the auxiliary variable in (6). This gap is load-bearing: it feeds Theorem IV.2 and, through Theorem V.1, Theorem III.1.
  2. [VI, Lemma VI.3 and Theorem VI.2] The same structural gap reappears in the k-dimensional case. Lemma VI.3 proves concentration of the idealized sum sum_{i=1}^{tilde-n} Z_i of coset parities, but phi(x) equals this sum only on the event G_k that all first-order projections at the leaves are decoded to zero. The passage from Lemma VI.3 to the concentration of phi(x), described as 'arguing similar to Lemma IV.4', is not justified off G_k. Consequently the bound in Theorem VI.2, and hence Theorem III.2, inherits the unproved step.
minor comments (5)
  1. [III, before Theorem III.1] The symbol p is used both for the channel crossover probability and for the projected noise level (1-(1-2p)^{2^{r-2}})/2; this is confusing and should be resolved with a distinct symbol such as bar-p.
  2. [IV-B, aggregation step] The notation for the normalized count is circular: the text writes phi(x) = phi^(N)(x) = phi(x)/(N-1). The normalized quantity should be introduced as a separate symbol, e.g., bar-phi(x).
  3. [IV-B, proof of Proposition IV.2] The event G is reused for two different events in the same proof (the event that all projections decode to zero and the concentration event for the normalized count). Renaming one of them would remove ambiguity.
  4. [IV-B, Lemma IV.5] The appeal to '[33, Lemma 1]' is too vague; either state the lemma or provide the short argument, since the cited paper is about coset error patterns and is not a standard background reference.
  5. [Throughout] There are a few typos, including 'probabailities' in Section III and 'straightfoward' in Section VI; a careful proofreading pass is needed.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the error-probability bounds are derived from independent concentration inequalities and prior external results, not from the claims being proved.

full rationale

I traced the main chain Theorem III.1 -> Theorem V.1 -> Theorem IV.2 -> Proposition IV.2 -> Theorem IV.1 plus Lemmas IV.2-IV.4. Every bound is a genuine upper bound computed from Hoeffding's inequality or the Lipschitz concentration theorem; epsilon is an arbitrary parameter in a stated range and no parameter is fitted to the target error probability. The reduction to the all-zeros codeword is imported from [23, Prop. 2], prior external work by Ye and Abbe, and it is not equivalent to the RPA error bound being derived. The RPA algorithm itself is the object under study, taken from [23] as an external definition rather than cited as evidence for its own performance. The recursive projection-aggregation analysis conditions on the event that all child decodes are correct and then adds the complement probability by a union bound, which is a standard decomposition rather than a definitional identity. I therefore find no circular dependency. One non-circular correctness concern: Lemma IV.4 is stated unconditionally, but its proof uses identity (6), which is justified only on the event that every FHT projection is decoded to the all-zero word; the off-event probability is not carried through. This is a missing argument that would need repair, but it does not make Theorem III.1 an input to itself or turn any prediction into a fit.

Assumptions & free parameters 2 free parameters · 4 assumptions · 0 invented entities

The central claim rests on standard concentration inequalities, standard RM code facts, and one imported decoder-symmetry result [23, Prop. 2]. No data fitting occurs; epsilon and delta are analytic slack parameters. The only non-elementary ingredient is the Lipschitz concentration bound for the k>1 analysis.

free parameters (2)
  • epsilon (epsilon) = arbitrary in (0, eta(p))
    A slack parameter in the concentration bounds; the theorems hold uniformly for any such epsilon, and corollaries optimize over epsilon via epsilon^2 = delta eta(p)^2.
  • delta (delta) = arbitrary in (0,1)
    Chosen when converting the epsilon-based bound into an explicit exponent in Corollaries III.1, III.2, VI.1; not fitted to data.
assumptions (4)
  • standard math Standard Hoeffding inequality (and its two-sided form) for sums of independent bounded random variables
    Used in Lemmas IV.2, IV.3, IV.4.
  • standard math Concentration inequality for Lipschitz functions of i.i.d. Bernoulli variables (Raginsky-Sason, Thm. 3.4.4, cited as [34])
    Used in Lemma VI.3 for the k-dimensional subspace analysis.
  • domain assumption RM code properties: dimension, minimum distance 2^{m-r}, and closure of projections onto cosets: c/B_i in RM(m-k,r-k) ([23, Lemma 1])
    Invoked in Section II and throughout the recursion; standard facts about Reed-Muller codes.
  • domain assumption Codeword symmetry of RPA error probability over BSC, [23, Prop. 2]
    Imported in Section III to restrict to the all-zeros codeword; not re-proven in this paper.

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Cite this review

Pith. "Pith review of An Analysis of RPA Decoding of Reed-Muller Codes Over the BSC." pith.science (2026). https://pith.science/paper/ESCCMJUA

@misc{pith2026241208129,
  author       = {Pith},
  title        = {Pith review of: An Analysis of RPA Decoding of Reed-Muller Codes Over the BSC},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/ESCCMJUA}},
  note         = {Machine review of arXiv:2412.08129}
}
abstract

In this paper, we revisit the Recursive Projection-Aggregation (RPA) decoder, of Ye and Abbe (2020), for Reed-Muller (RM) codes. Our main contribution is an explicit upper bound on the probability of incorrect decoding, using the RPA decoder, over a binary symmetric channel (BSC). Importantly, we focus on the events where a \emph{single} iteration of the RPA decoder, in each recursive call, is sufficient for convergence. Key components of our analysis are explicit estimates of the probability of incorrect decoding of first-order RM codes using a maximum likelihood (ML) decoder, and estimates of the error probabilities during the aggregation phase of the RPA decoder. Our results allow us to show that for RM codes with blocklength $N = 2^m$, the RPA decoder can achieve vanishing error probabilities, in the large blocklength limit, for RM orders that grow roughly logarithmically in $m$.

Figures

Figures reproduced from arXiv: 2412.08129 by the authors.

Figure 1
Figure 1. Figure representing a projection-aggregation tree [PITH_FULL_IMAGE:figures/full_fig_p017_1.png] view at source ↗
Figure 2
Figure 2. Figure representing a projection-aggregation tree using subspaces of dimension [PITH_FULL_IMAGE:figures/full_fig_p022_2.png] view at source ↗

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Forward citations

Cited by 2 Pith papers

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    The paper proves that the Gibbs decoder for Reed-Muller codes can have super-polynomial mixing time on certain typical received sequences, so it is not a worst-case polynomial-time decoder.

  2. Capacity on BMS Channels via Code Symmetry and Nesting

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    The paper derives recursive two-look and three-look bounds showing Reed-Muller codes achieve capacity on BMS channels and a faster BSC decay rate via level-k inequalities.

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