[Paper Review] Pruned Collapsed Projection-Aggregation Decoding of Reed-Muller Codes

Qin Huang, Bin Zhang|arXiv (Cornell University)|May 25, 2021

Error Correcting Code Techniques9 references4 citations

TL;DR

This paper proposes Pruned Collapsed Projection-Aggregation (PCPA) decoding for Reed-Muller codes by selecting a small subset of subspaces for projection based on a correlation coefficient that measures subspace intersection size. By favoring subspaces with low correlation coefficients, PCPA achieves performance close to full CPA decoding with significantly reduced complexity, outperforming RPA and CPA in complexity while maintaining near-optimal error correction for RM(3,7) and RM(3,5) codes.

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.

Motivation & Objective

• To reduce the computational complexity of collapsed projection-aggregation (CPA) decoding for Reed-Muller codes by limiting the number of subspaces used in projection.
• To identify a criterion for selecting informative subspaces that maximizes decoding performance with minimal projections.
• To develop a correlation coefficient metric based on subspace intersection size to guide subspace selection.
• To demonstrate that low-correlation subspace collections yield better error performance than high-correlation ones.

Proposed method

• Proposes a pruned collapsed projection-aggregation (PCPA) algorithm that projects the received vector onto a selected subset of (r−1)-dimensional subspaces instead of all N_{r,m} subspaces.
• Introduces a correlation coefficient r_{ij} = dim(𝐵_i ∩ 𝐵_j)/s to quantify the intersection size between two s-dimensional subspaces 𝐵_i and 𝐵_j.
• Defines the total correlation coefficient r_S = ΣᵢΣⱼ r_ij for a collection S of subspaces to evaluate overall subspace redundancy.
• Uses Monte Carlo simulations to show that subspaces with lower r_S yield higher agreement in estimates and better decoding performance.
• Employs majority voting across estimates from selected subspaces to iteratively refine the log-likelihood ratio vector.
• Applies fast Hadamard transform (FHT) to decode the projected RM(1, m−r+1) codes obtained from each subspace projection.

Experimental results

Research questions

• RQ1 Can a small, carefully selected subset of subspaces achieve near-optimal decoding performance in RM code decoding with significantly reduced complexity?
• RQ2 How does the intersection size between subspaces affect the likelihood of error cancellation and estimate agreement in projection-aggregation decoding?
• RQ3 Can a correlation coefficient based on subspace intersection size serve as an effective metric for subspace selection in collapsed projection-aggregation decoding?
• RQ4 Does selecting subspaces with low correlation coefficients lead to better error performance compared to random or high-correlation selections?
• RQ5 How does the performance of PCPA compare to CPA and RPA in terms of word error rate and computational complexity for RM(3,7) and RM(3,5) codes?

Key findings

• PCPA with 64 projections for RM(3,7) achieves a word error rate (WER) of 10⁻³ at only 0.1 dB degradation compared to CPA and RPA.
• PCPA with 64 projections reduces computational complexity from 2667 projections (CPA) to 64, achieving a 97.8% reduction in projection count.
• For RM(3,5), PCPA with 9 projections and r_S = 9 outperforms a collection with r_S = 31, demonstrating that lower correlation coefficients yield better performance.
• The performance gap between PCPA and CPA is only 0.25 dB at WER = 10⁻³ when comparing collections with r_S = 69 and r_S = 313 for RM(3,7).
• The correlation coefficient r_S effectively predicts subspace collection quality: lower r_S correlates with higher estimate agreement and better decoding performance.
• PCPA with 9 projections for RM(3,5) achieves performance very close to CPA and the lower union bound, validating the effectiveness of the correlation-based selection strategy.