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Subcode Ensemble Decoding of Polar Codes

Henning Lulei, Jonathan Mandelbaum, Marvin Rübenacke, Holger Jäkel, Stephan ten Brink, Laurent Schmalen

TL;DR

Paper addresses the challenge of improving short-block polar codes under fixed hardware area by augmenting SC-based decoding with Subcode Ensemble Decoding (ScED). It introduces a unified pre-transformations (PT) framework that classifies PTs into PT-A, PT-B, and PT-C to generate polar subcodes and enable decoding on a joint graph; PT-C is used exclusively in decoding to realize subcode ensembles that cover the code. The authors show how to optimize depth $d_p$ and target bits and propose an ensemble design procedure that selects $M$ PT-C paths from $r$ candidates to maximize coverage while maintaining path-metric ordering. Empirical results on $\mathcal{C}_T(64,32)$ and $\mathcal{C}_T(256,k)$ demonstrate SCED-$M$-SCL-$L$ surpassing stand-alone SCL decoders at the same $L$ by about $0.1$–$0.25$ dB, with the short code achieving the performance of SCL-$2L$ using only $M=2$ paths. This approach offers hardware-efficient routes to improved polar decoding, enabling promising gains in short-block regimes and potential latency reductions when area constraints are relaxed.

Abstract

In the short block length regime, pre-transformed polar codes together with successive cancellation list (SCL) decoding possess excellent error correction capabilities. However, in practice, the list size is limited due to the suboptimal scaling of the required area in hardware implementations. Automorphism ensemble decoding (AED) can improve performance for a fixed list size by running multiple parallel SCL decodings on permuted received words, yielding a list of estimates from which the final estimate is selected. Yet, AED is limited to appropriately designed polar codes. Subcode ensemble decoding (ScED) was recently proposed for low-density parity-check codes and does not impose such design constraints. It uses multiple decodings in different subcodes, ensuring that the selected subcodes jointly cover the original code. We extend ScED to polar codes by expressing polar subcodes through suitable pre-transformations (PTs). To this end, we describe a framework classifying pre-transformations for pre-transformed polar codes based on their role in encoding and decoding. Within this framework, we propose a new type of PT enabling ScED for polar codes, analyze its properties, and discuss how to construct an efficient ensemble.

Subcode Ensemble Decoding of Polar Codes

TL;DR

Paper addresses the challenge of improving short-block polar codes under fixed hardware area by augmenting SC-based decoding with Subcode Ensemble Decoding (ScED). It introduces a unified pre-transformations (PT) framework that classifies PTs into PT-A, PT-B, and PT-C to generate polar subcodes and enable decoding on a joint graph; PT-C is used exclusively in decoding to realize subcode ensembles that cover the code. The authors show how to optimize depth and target bits and propose an ensemble design procedure that selects PT-C paths from candidates to maximize coverage while maintaining path-metric ordering. Empirical results on and demonstrate SCED--SCL- surpassing stand-alone SCL decoders at the same by about – dB, with the short code achieving the performance of SCL- using only paths. This approach offers hardware-efficient routes to improved polar decoding, enabling promising gains in short-block regimes and potential latency reductions when area constraints are relaxed.

Abstract

In the short block length regime, pre-transformed polar codes together with successive cancellation list (SCL) decoding possess excellent error correction capabilities. However, in practice, the list size is limited due to the suboptimal scaling of the required area in hardware implementations. Automorphism ensemble decoding (AED) can improve performance for a fixed list size by running multiple parallel SCL decodings on permuted received words, yielding a list of estimates from which the final estimate is selected. Yet, AED is limited to appropriately designed polar codes. Subcode ensemble decoding (ScED) was recently proposed for low-density parity-check codes and does not impose such design constraints. It uses multiple decodings in different subcodes, ensuring that the selected subcodes jointly cover the original code. We extend ScED to polar codes by expressing polar subcodes through suitable pre-transformations (PTs). To this end, we describe a framework classifying pre-transformations for pre-transformed polar codes based on their role in encoding and decoding. Within this framework, we propose a new type of PT enabling ScED for polar codes, analyze its properties, and discuss how to construct an efficient ensemble.

Paper Structure

This paper contains 13 sections, 1 theorem, 4 equations, 7 figures, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Polar Codes
  4. Pre-transformed Polar Codes
  5. Subcodes and Subcode Ensemble Decoding (ScED)
  6. Subcode Ensemble Decoding of Polar Codes
  7. Generalized Framework for Pre-transformations
  8. Applying Pre-transformations in ScED
  9. Selection of Pre-transformations for ScED
  10. Parameter Analysis
  11. Design of an Effective Ensemble for ScED
  12. Results
  13. Conclusion

Key Result

Theorem 1

Consider a polar code $\mathcal{C}(N, k)$ with SC decoding $\mathrm{SC}: \mathcal{Y}^N \to \mathcal{C}$. Consider a PT-C~such that $\mathcal{C}_T \subseteq \mathcal{C}$ and let $\mathrm{SC}_T: \mathcal{Y}^N \to \mathcal{C}_T$ denote the decoder on the joint graph. Assume that codeword $\bm{x} \in \m i.e., if a stand-alone SC decoding decodes correctly, a subcode SC decoding will make the same deci

Figures (7)

  • Figure 1: Pre-transformed polar code formed by employing PT-A~and PT-B.
  • Figure 2: Exemplary joint encoding graph for the polar code $\mathcal{C}(8, 4)$ with information set ${\mathcal{I} = \{3, 5, 6, 7\}}$ and PT with $\mathcal{T}=\{4, 7\}$ from (\ref{['eq:PTexample']}), resulting in the pre-transformed polar code $\mathcal{C}_T(8,3)$. Information bits of $\mathcal{C}$ are colored in orange. Target bits $t \in \mathcal{T}$ of the PT are drawn as triangles (/ ).
  • Figure 3: SCED employing PT-B~and $M$PT-C s. PT-A s~are considered for the genie-aided decision.
  • Figure 4: Ratio of decoded URP over the indices of target bits of PT sorted by the reliability of the corresponding bit-channels for the DE polar code $\mathcal{C}(256,128)$ with SC decoding in ascending order. Analysis based on ${2000 \cdot 100}$URP using $192$ newly sampled subcode decoders per $100$URP.
  • Figure 5: Ratio of decoded URP over the depth $d_\mathrm{p}$ of generated PT. Analyzed $2000 \cdot 100$URP using $190$ new decoders referencing PT per $100$URP for the DE polar code $\mathcal{C}(256,128)$ with SC decoding.
  • ...and 2 more figures

Theorems & Definitions (2)

  • Theorem 1
  • proof