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Sparsely Pre-transformed Polar Codes for Low-Latency SCL Decoding

Geon Choi, Namyoon Lee

TL;DR

The paper tackles the challenge of achieving low-latency, finite-blocklength reliability in polar-encoded communications by improving pre-transformed polar codes for small-list SCL decoding. It introduces sparsely pre-transformed polar (SPP) codes that run two parallel pre-transform types—Type-i and Type-ii—to simultaneously mitigate two distinct SCL error events, $P_{ m SCL}({\mathcal E}_1;S)$ and $P_{ m SCL}({\mathcal E}_2;S)$. The approach combines rate-profile optimization, bit-swapping, and upper-triangular pre-transform ideas, with encoding implemented through a sparse, block-diagonal pre-transform matrix ${\bf T}$ and parallel pre-transform blocks. Extensive simulations show that SPP codes outperform state-of-the-art pre-transformed polar codes (CRC-aided polar, PAC, and deep polar) under small list sizes, especially at short blocklengths, highlighting their potential for URLLC-like latency reductions. The work thus provides a practical design framework for high-performance, low-latency polar codes that can be integrated with CRC precoding to further improve distance properties and decoding robustness.

Abstract

Deep polar codes, employing multi-layered polar kernel pre-transforms in series, are recently introduced variants of pre-transformed polar codes. These codes have demonstrated the ability to reduce the number of minimum weight codewords, thereby closely achieving finite-block length capacity with successive cancellation list (SCL) decoders in certain scenarios. However, when the list size of the SCL decoder is small, which is crucial for low-latency communication applications, the reduction in the number of minimum weight codewords does not necessarily improve decoding performance. To address this limitation, we propose an alternative pre-transform technique to enhance the suitability of polar codes for SCL decoders with practical list sizes. Leveraging the fact that the SCL decoding error event set can be decomposed into two exclusive error event sets, our approach applies two different types of pre-transformations, each targeting the reduction of one of the two error event sets. Extensive simulation results under various block lengths and code rates have demonstrated that our codes consistently outperform all existing state-of-the-art pre-transformed polar codes, including CRC-aided polar codes and polarization-adjusted convolutional codes, when decoded using SCL decoders with small list sizes.

Sparsely Pre-transformed Polar Codes for Low-Latency SCL Decoding

TL;DR

The paper tackles the challenge of achieving low-latency, finite-blocklength reliability in polar-encoded communications by improving pre-transformed polar codes for small-list SCL decoding. It introduces sparsely pre-transformed polar (SPP) codes that run two parallel pre-transform types—Type-i and Type-ii—to simultaneously mitigate two distinct SCL error events, and . The approach combines rate-profile optimization, bit-swapping, and upper-triangular pre-transform ideas, with encoding implemented through a sparse, block-diagonal pre-transform matrix and parallel pre-transform blocks. Extensive simulations show that SPP codes outperform state-of-the-art pre-transformed polar codes (CRC-aided polar, PAC, and deep polar) under small list sizes, especially at short blocklengths, highlighting their potential for URLLC-like latency reductions. The work thus provides a practical design framework for high-performance, low-latency polar codes that can be integrated with CRC precoding to further improve distance properties and decoding robustness.

Abstract

Deep polar codes, employing multi-layered polar kernel pre-transforms in series, are recently introduced variants of pre-transformed polar codes. These codes have demonstrated the ability to reduce the number of minimum weight codewords, thereby closely achieving finite-block length capacity with successive cancellation list (SCL) decoders in certain scenarios. However, when the list size of the SCL decoder is small, which is crucial for low-latency communication applications, the reduction in the number of minimum weight codewords does not necessarily improve decoding performance. To address this limitation, we propose an alternative pre-transform technique to enhance the suitability of polar codes for SCL decoders with practical list sizes. Leveraging the fact that the SCL decoding error event set can be decomposed into two exclusive error event sets, our approach applies two different types of pre-transformations, each targeting the reduction of one of the two error event sets. Extensive simulation results under various block lengths and code rates have demonstrated that our codes consistently outperform all existing state-of-the-art pre-transformed polar codes, including CRC-aided polar codes and polarization-adjusted convolutional codes, when decoded using SCL decoders with small list sizes.
Paper Structure (25 sections, 2 theorems, 49 equations, 7 figures, 3 tables, 3 algorithms)

This paper contains 25 sections, 2 theorems, 49 equations, 7 figures, 3 tables, 3 algorithms.

Table of Contents

  1. Introduction
  2. Related Works
  3. Contributions
  4. Preliminaries
  5. Channel Coding System
  6. Polar Codes
  7. Pre-Transformed Polar Codes
  8. SCL Decoding for Pretransformed Polar Codes
  9. Notations
  10. SCL Decoding Error Analysis
  11. Two Error Events in SCL Decoding
  12. Rate Profiling Method to Reduce $P_{\sf SCL} ({\mathcal{E}}_1; S)$
  13. Bit-Swapping Method to Reduce $P_{\sf SCL} ({\mathcal{E}}_2; S)$
  14. Bit-Swapping to Reduce both $P_{\sf SCL}({\mathcal{E}}_1; S)$ and $P_{\sf SCL}({\mathcal{E}}_2; S)$
  15. Upper-Triangular Pre-Transform to Reduce $P_{\sf SCL}({\mathcal{E}}_2; S)$
  16. ...and 10 more sections

Key Result

Lemma 1

The incremental entropy by decoding $u_m$ is

Figures (7)

  • Figure 1: Decoding of pre-transformed polar codes
  • Figure 2: Illustration of Encoding Structure of SPP codes. Type-i pre-transforms are marked in orange, and Type-ii pre-transforms are marked in green.
  • Figure 3: (Example 1) Bit channel capacity (left) and normalized row weight (right) on binary erasure channel (BEC) with erasure probability of $1/2$.
  • Figure 4: (Example 3) Path metric range evaluated with list size of $S=2$ averaged over $10^3$ channel noises generated at ${E_b/N_0} = 3$ dB. Top: $(N,K) = (128,32)$, Bottom: $(N,K)=(128,96)$.
  • Figure 5: (Example 4) Bit channel capacity (left) and normalized row weight (right) on binary erasure channel (BEC) with erasure probability of 1/2.
  • ...and 2 more figures

Theorems & Definitions (2)

  • Lemma 1
  • Theorem 1