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Subcode Ensemble Decoding of Linear Block Codes

Jonathan Mandelbaum, Holger Jäkel, Laurent Schmalen

TL;DR

This work tackles the short-block LDPC decoding gap between BP and ML decoding by introducing Subcode Ensemble Decoding (SCED), which runs parallel BP decoders on subcodes induced by appended rows to the parity-check matrix. It leverages linear coverings to ensure all codewords are decodable and uses a maximum-coverage heuristic to select effective subcodes, achieving gains over stand-alone BP and existing ensemble methods without requiring NP-hard dual-codeword searches or automorphism group knowledge. Across simulations on short LDPC codes, SCED delivers FER improvements and lower worst-case latency, while remaining broadly applicable and easy to construct. The approach holds promise for ultra-reliable low-latency communications and can be extended to other code families.

Abstract

Low-density parity-check (LDPC) codes together with belief propagation (BP) decoding yield exceptional error correction capabilities in the large block length regime. Yet, there remains a gap between BP decoding and maximum likelihood decoding for short block length LDPC codes. In this context, ensemble decoding schemes yield both reduced latency and good error rates. In this paper, we propose subcode ensemble decoding (SCED), which employs an ensemble of decodings on different subcodes of the code. To ensure that all codewords are decodable, we use the concept of linear coverings and explore approaches for sampling suitable ensembles for short block length LDPC codes. Monte-Carlo simulations conducted for three LDPC codes demonstrate that SCED improves decoding performance compared to stand-alone decoding and automorphism ensemble decoding. In particular, in contrast to existing schemes, e.g., multiple bases belief propagation and automorphism ensemble decoding, SCED does not require the NP-complete search for low-weight dual codewords or knowledge of the automorphism group of the code, which is often unknown.

Subcode Ensemble Decoding of Linear Block Codes

TL;DR

This work tackles the short-block LDPC decoding gap between BP and ML decoding by introducing Subcode Ensemble Decoding (SCED), which runs parallel BP decoders on subcodes induced by appended rows to the parity-check matrix. It leverages linear coverings to ensure all codewords are decodable and uses a maximum-coverage heuristic to select effective subcodes, achieving gains over stand-alone BP and existing ensemble methods without requiring NP-hard dual-codeword searches or automorphism group knowledge. Across simulations on short LDPC codes, SCED delivers FER improvements and lower worst-case latency, while remaining broadly applicable and easy to construct. The approach holds promise for ultra-reliable low-latency communications and can be extended to other code families.

Abstract

Low-density parity-check (LDPC) codes together with belief propagation (BP) decoding yield exceptional error correction capabilities in the large block length regime. Yet, there remains a gap between BP decoding and maximum likelihood decoding for short block length LDPC codes. In this context, ensemble decoding schemes yield both reduced latency and good error rates. In this paper, we propose subcode ensemble decoding (SCED), which employs an ensemble of decodings on different subcodes of the code. To ensure that all codewords are decodable, we use the concept of linear coverings and explore approaches for sampling suitable ensembles for short block length LDPC codes. Monte-Carlo simulations conducted for three LDPC codes demonstrate that SCED improves decoding performance compared to stand-alone decoding and automorphism ensemble decoding. In particular, in contrast to existing schemes, e.g., multiple bases belief propagation and automorphism ensemble decoding, SCED does not require the NP-complete search for low-weight dual codewords or knowledge of the automorphism group of the code, which is often unknown.
Paper Structure (14 sections, 2 theorems, 16 equations, 5 figures, 1 table, 1 algorithm)

This paper contains 14 sections, 2 theorems, 16 equations, 5 figures, 1 table, 1 algorithm.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Subcode Ensemble Decoding
  4. Approaches to Choose Suitable Subcodes
  5. Suitable Subcodes for BP Decoding
  6. Linear Covering of Codes
  7. Maximum-Coverage Heuristic
  8. Comparison
  9. Comparison to Existing Schemes
  10. Results
  11. Coverage Characteristics
  12. Frame Error Rate Results
  13. Average Latency & Qualitative Results
  14. Conclusion

Key Result

Theorem 1

Let ${\bm{h}_1,\bm{h}_2\in \mathbb{F}^{1\times n}}$ be two row vectors that are linearly independent of the rows of a PCM $\bm{H}$ of $\mathcal{C}$ and let ${\bm{h}_3=\bm{h}_2+\bm{h}_1}$. Using (eq:inducing_subcode), they induce subcodes ${\mathcal{C}_1,\mathcal{C}_2\subset \mathcal{C}},$ and $\math

Figures (5)

  • Figure 1: Block diagram of SCED using $K$ different subcodes $\mathcal{C}_i\subseteq \mathcal{C}$ and their respective decoder $\mathrm{Dec}_i$.
  • Figure 2: Performance of SCED using the different ensembles and stand-alone decoding for the 5G LDPC code $\mathcal{C}_{5\mathrm{G}}(132,66)$.
  • Figure 3: Relative coverage as a function of the number of additional paths $\Tilde{K}$
  • Figure 4: Decoder performances for the 5G LDPC code $\mathcal{C}_{5\mathrm{G}}(132,66)$.
  • Figure 5: Decoder performances for the code $\mathcal{C}_{\mathrm{irPEG}}(504,252)$ from mackay_codes_files.

Theorems & Definitions (4)

  • Theorem 1
  • Lemma 1
  • proof : Proof of Lemma \ref{['lemma:proof_of_existence']}
  • proof : Proof of Theorem \ref{['theorem:lc_k-1_dim']}