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Hierarchical Subcode Ensemble Decoding of Polar Codes

Yubeen Jo, Geon Choi, Chanho Park, Namyoon Lee

TL;DR

This work tackles the BP decoding bottleneck for polar codes under URLLC constraints by introducing Hierarchical Subcode Ensemble Decoding (HSCED). HSCED builds a tree-structured ensemble of subcodes through recursive, linearly independent parity-constraint augmentation on a sparse, RREF-derived base graph, ensuring a guaranteed linear covering property with a growth of the ensemble size as $3^m$. The framework enables $3^m+1$ parallel BP decoders and an ML-in-the-list selection, yielding significant block-error-rate gains over standard BP and conventional SCED, while maintaining bounded latency suitable for latency-critical applications. Empirical results on 5G NR polar codes show that HSCED narrows the gap to SCL-based decoding in reliable regimes and offers a favorable complexity-latency trade-off for URLLC scenarios, driven by systematic stopping-set reduction and controlled graph sparsity. Overall, HSCED provides a scalable, high-throughput approach to near-ML reliability for polar codes with fixed latency guarantees, leveraging structured graph diversity and parallelism.

Abstract

Subcode-ensemble decoders improve iterative decoding by running multiple decoders in parallel over carefully chosen subcodes, increasing the likelihood that at least one decoder avoids the dominant trapping structures. Achieving strong diversity gains, however, requires constructing many subcodes that satisfy a linear covering property-yet existing approaches lack a systematic way to scale the ensemble size while preserving this property. This paper introduces hierarchical subcode ensemble decoding (HSCED), a new ensemble decoding framework that expands the number of constituent decoders while still guaranteeing linear covering. The key idea is to recursively generate subcode parity constraints in a hierarchical structure so that coverage is maintained at every level, enabling large ensembles with controlled complexity. To demonstrate its effectiveness, we apply HSCED to belief propagation (BP) decoding of polar codes, where dense parity-check matrices induce severe stopping-set effects that limit conventional BP. Simulations confirm that HSCED delivers significant block-error-rate improvements over standard BP and conventional subcode-ensemble decoding under the same decoding-latency constraint.

Hierarchical Subcode Ensemble Decoding of Polar Codes

TL;DR

This work tackles the BP decoding bottleneck for polar codes under URLLC constraints by introducing Hierarchical Subcode Ensemble Decoding (HSCED). HSCED builds a tree-structured ensemble of subcodes through recursive, linearly independent parity-constraint augmentation on a sparse, RREF-derived base graph, ensuring a guaranteed linear covering property with a growth of the ensemble size as . The framework enables parallel BP decoders and an ML-in-the-list selection, yielding significant block-error-rate gains over standard BP and conventional SCED, while maintaining bounded latency suitable for latency-critical applications. Empirical results on 5G NR polar codes show that HSCED narrows the gap to SCL-based decoding in reliable regimes and offers a favorable complexity-latency trade-off for URLLC scenarios, driven by systematic stopping-set reduction and controlled graph sparsity. Overall, HSCED provides a scalable, high-throughput approach to near-ML reliability for polar codes with fixed latency guarantees, leveraging structured graph diversity and parallelism.

Abstract

Subcode-ensemble decoders improve iterative decoding by running multiple decoders in parallel over carefully chosen subcodes, increasing the likelihood that at least one decoder avoids the dominant trapping structures. Achieving strong diversity gains, however, requires constructing many subcodes that satisfy a linear covering property-yet existing approaches lack a systematic way to scale the ensemble size while preserving this property. This paper introduces hierarchical subcode ensemble decoding (HSCED), a new ensemble decoding framework that expands the number of constituent decoders while still guaranteeing linear covering. The key idea is to recursively generate subcode parity constraints in a hierarchical structure so that coverage is maintained at every level, enabling large ensembles with controlled complexity. To demonstrate its effectiveness, we apply HSCED to belief propagation (BP) decoding of polar codes, where dense parity-check matrices induce severe stopping-set effects that limit conventional BP. Simulations confirm that HSCED delivers significant block-error-rate improvements over standard BP and conventional subcode-ensemble decoding under the same decoding-latency constraint.
Paper Structure (14 sections, 1 theorem, 3 equations, 7 figures, 4 tables, 1 algorithm)

This paper contains 14 sections, 1 theorem, 3 equations, 7 figures, 4 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Channel Coding and Polar Codes
  4. BP Decoding and Structural Weaknesses
  5. Subcode Ensemble Decoding
  6. Hierarchical Subcode Ensemble Decoding
  7. Recursive Expansion via Linear Covering
  8. Parallel BP Ensemble Decoding
  9. Application to Polar Codes
  10. Structural Effects of HSCED
  11. Simulation Results
  12. Simulation Settings and Benchmarks
  13. Performance, Complexity, and Latency Analysis
  14. Conclusion

Key Result

Theorem 1

Let $\mathcal{C} = \text{Null}(\boldsymbol{H})$. The union of subcodes defined by the ensemble ${\mathcal{H}}_m$ of $3^m$ PCMs (indexed by $\mathbf{i} \in \{1,2,3\}^m$) forms a complete linear covering of $\mathcal{C}$:

Figures (7)

  • Figure 1: Illustration of structural weaknesses in PCM $\boldsymbol{H}$ where a 4-cycle (blue) represents the smallest stopping set and a larger stopping set is shown in red.
  • Figure 2: Hierarchical structure of the proposed HSCED ensemble. $\boldsymbol{H}_{i_1,\dots,i_m}$ denotes the PCM of a subcode generated by augmenting the base PCM with row vectors.
  • Figure 3: Architecture of the proposed HSCED decoder. At depth $m$, a total of $3^m+1$ parallel BP decoders operate on distinct subcodes.
  • Figure 4: Heatmap comparison of the polar (64, 32) PCM before ($\boldsymbol{H}$) and after RREF application ($\tilde{\boldsymbol{H}}$).
  • Figure 5: Asymmetry of gain (stopping set reduction) versus cost (4-cycle increase) when adding rows to the RREF-Polar $(64, 32)$ graph $T(\tilde{\boldsymbol{H}})$.
  • ...and 2 more figures

Theorems & Definitions (1)

  • Theorem 1: Hierarchical Linear Covering