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IEEE Transactions on Information Theory

Coset Error Pattern in Projection-Aggregation Decoding

Bin Zhang, Fanyun Chen, Qin Huang

Published Jun 5, 2025• 3 citations
Publisher

Abstract

Projection-aggregation decoding provides state-of-the-art performance for decoding Reed-Muller codes. This type of decoding relies on projecting onto subspaces, decoding the projections, and voting with the decoded projections. This paper investigates the decoding with a limited number of subspaces. By defining a false vote matrix, it demonstrates that this decoding suffers from coset error patterns, even when all projections are decoded successfully. These error patterns are of very small weights and thus induce a significant performance loss. Two rules are then proposed of which one reveals a trade-off between the subspace number and the error-correcting capability and the other provides a guideline for subspace selection. Simulation results verify that following the two rules alleviates the performance loss caused by the coset error patterns.

DOI
10.1109/tit.2025.3576779

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Projection-aggregation decoding provides state-of-the-art performance for decoding Reed-Muller codes. This type of decoding relies on projecting onto subspaces, decoding the projections, and voting with the decoded projections. This paper investigates the decoding with a limited number of subspaces. By defining a false vote matrix, it demonstrates that this decoding suffers from coset error patterns, even when all projections are decoded successfully. These error patterns are of very small weights and thus induce a significant performance loss. Two rules are then proposed of which one reveals a trade-off between the subspace number and the error-correcting capability and the other provides a guideline for subspace selection. Simulation results verify that following the two rules alleviates the performance loss caused by the coset error patterns.
The recursive projection-aggregation (RPA) algorithm provides state-of-the-art performance for Reed-Muller (RM) codes. It relies on projecting onto subspaces, decoding projections and voting. This paper explores the error-correction capability of RPA with a limited number of subspaces. By constructing a false vote matrix, it demonstrates that RPA may fail to correct coset error patterns, even when all projections are decoded successfully. These patterns severely degrade the decoding performance due to their small weights. Thus, two rules on the subspace selection are derived to guarantee the error-correction capability of RPA. Simulation results verify that the two rules dominate the decoding performance of RPA.
This paper shows that projection-aggregation (PA) decoding of Reed-Muller (RM) codes can decode up to half the minimum distance d efficiently. By generalizing the false vote matrix from 1-dimensional to higher-dimensional subspaces, we prove that projecting onto disjoint subspaces, regardless of their dimensions, provides at most d/2 − 1 false votes for each codeword bit. Thus, d disjoint subspaces guarantee the correction for errors of d/2 − 1 or less. Moreover, the flexibility in subspace dimensions allows projecting into repetition codes directly, resulting in decoding efficiency. Finally, we prove that PA decoding with d disjoint subspaces decodes up to half the minimum distance in O(nn)O(n\sqrt n )O(nn​) for RM codes of length n and half rate or less.
The paper proposes to decode Reed-Muller (RM) codes by projecting onto only a few subspaces such that the number of projections is significantly reduced. It reveals that the probability that error pairs are canceled simultaneously in two different projections is determined by their intersection size. Then, correlation coefficient which indicates the intersection size of two subspaces in a collection is introduced for collecting subspaces. Simulation results show that our proposed approach with a small number of projections onto collected subspaces performs close to the original approach.
Recursive projection aggregation (RPA) decoding as introduced in [1] is a novel decoding algorithm which performs close to the maximum likelihood decoder for short-length Reed-Muller codes. Recently, an extension to RPA decoding, called sparse multi-decoder RPA (SRPA), has been proposed [2]. The SRPA approach makes use of multiple pruned RPA decoders to lower the amount of computations while keeping the performance loss small compared to RPA decoding. However, the use of multiple sparse decoders again increases the computational burden. Therefore, the focus is on the optimization of sparse single-decoder RPA decoding to keep the complexity small. In this paper, a novel method is proposed, to select subsets of subspaces used in the projection and aggregation step of SRPA decoding in order to decrease the decoding error probability on AWGN channels. The proposed method replaces the random selection of subspace subsets with a semi-deterministic selection method based on a figure of merit that evaluates the performance of each subspace. Our simulation results show that the semi-deterministic subspace selection improves the decoding performance up to 0.2 dB0.2\,\text{dB}0.2dB compared to SRPA. At the same time, the complexity of SRPA decoding for RM codes of order r≥3r\geq 3r≥3 is reduced by up to 81% compared to SRPA.
We propose a new class of efficient decoding algorithms for Reed-Muller (RM) codes over binary-input memoryless channels. The algorithms are based on projecting the code on its cosets, recursively decoding the projected codes (which are lower-order RM codes), and aggregating the reconstructions (e.g., using majority votes). We further provide extensions of the algorithms using list-decoding. We run our algorithm for AWGN channels and Binary Symmetric Channels at the short code length (≤ 1024) regime for a wide range of code rates. Simulation results show that in both low code rate and high code rate regimes, the new algorithm outperforms the widely used decoder for polar codes (SCL+CRC) with the same parameters. The performance of the new algorithm for RM codes in those regimes is in fact close to that of the maximal likelihood decoder. Finally, the new decoder naturally allows for parallel implementations.