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arXiv:2105.11878

Pruned Collapsed Projection-Aggregation Decoding of Reed-Muller Codes

Qin Huang, Bin Zhang

Published May 25, 2021• 9 citations
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Abstract

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.

DOI
10.48550/arxiv.2105.11878

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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.
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 recursive projection-aggregation (RPA) decoder achieves near-maximum likelihood (ML) performance for lowrate Reed-Muller (RM) codes but suffers from high computational complexity when applied to high-rate codes. To address this limitation, we propose a structured subspace projection framework tailored to the code order: specifically, 2-dimensional subspaces for third-order RM codes. We demonstrate that decoding via projections onto these carefully chosen subspaces achieves better performance compared to an equal number of projections onto randomly selected subspaces. Further, we integrate automorphism ensemble decoding (AED) with our pruned projections, leveraging the rich symmetry group of RM codes to recover performance degradation from subspace pruning.
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.
Reed-Muller (RM) codes are one of the oldest families of codes. Recently, a recursive projection aggregation (RPA) decoder has been proposed, which achieves a performance that is close to the maximum likelihood decoder for short-length RM codes. One of its main drawbacks, however, is the large amount of computations needed. In this paper, we devise a new algorithm to lower the computational budget while keeping a performance close to that of the RPA decoder. The proposed approach consists of multiple sparse RPAs that are generated by performing only a selection of projections in each sparsified decoder. In the end, a cyclic redundancy check (CRC) is used to decide between output codewords. Simulation results show that our proposed approach reduces the RPA decoder's computations by up to 80% with negligible performance loss.