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PAC Codes Meet CRC-Polar Codes

Xinyi Gu, Mohammad Rowshan, Jinhong Yuan

TL;DR

The paper addresses finite-length performance gaps in polar-based codes by analyzing minimum weight codewords (MWCs) and exploiting coset structure. It introduces two schemes: Profile-Shifted PAC (PS-PAC) to reserve large-index rate-profile coordinates and reduce MWCs, and Continuous CRC-Polar (CCRC-polar) to insert CRC remainders continuously as parity checks during encoding and decoding. PS-PAC yields up to $0.5$ dB gains for short codes, and CCRC-polar delivers up to $0.12$ dB gains for long codes, with CRC-polar baselines also benefiting from MWC reductions. Together, these methods improve error performance under list decoding for a range of code lengths and rates, while maintaining practical complexity through precoding-inspired transformations.

Abstract

CRC-Polar codes under SC list decoding are well-regarded for their competitive error performance. This paper examines these codes by focusing on minimum weight codewords, breaking them down into the rows of the polar transform. Inspired by the significant impact of parity check bits and their positions, we apply a shifted rate-profile for polarization-adjusted convolutional (PS-PAC) codes, thereby achieving similar improvements in the weight distribution of polar codes through precoding. The results demonstrate a significant improvement in error performance, achieving up to a 0.5 dB power gain with short PS-PAC codes. Additionally, leveraging convolutional precoding in PAC codes, we adopt a continuous deployment (masking) of parity check bits derived from the remainder of continuous division of the partial message polynomial and the CRC polynomial over frozen positions in the rate-profile. This approach enhances performance for medium-length codes, with an overall improvement of 0.12 dB.

PAC Codes Meet CRC-Polar Codes

TL;DR

The paper addresses finite-length performance gaps in polar-based codes by analyzing minimum weight codewords (MWCs) and exploiting coset structure. It introduces two schemes: Profile-Shifted PAC (PS-PAC) to reserve large-index rate-profile coordinates and reduce MWCs, and Continuous CRC-Polar (CCRC-polar) to insert CRC remainders continuously as parity checks during encoding and decoding. PS-PAC yields up to dB gains for short codes, and CCRC-polar delivers up to dB gains for long codes, with CRC-polar baselines also benefiting from MWC reductions. Together, these methods improve error performance under list decoding for a range of code lengths and rates, while maintaining practical complexity through precoding-inspired transformations.

Abstract

CRC-Polar codes under SC list decoding are well-regarded for their competitive error performance. This paper examines these codes by focusing on minimum weight codewords, breaking them down into the rows of the polar transform. Inspired by the significant impact of parity check bits and their positions, we apply a shifted rate-profile for polarization-adjusted convolutional (PS-PAC) codes, thereby achieving similar improvements in the weight distribution of polar codes through precoding. The results demonstrate a significant improvement in error performance, achieving up to a 0.5 dB power gain with short PS-PAC codes. Additionally, leveraging convolutional precoding in PAC codes, we adopt a continuous deployment (masking) of parity check bits derived from the remainder of continuous division of the partial message polynomial and the CRC polynomial over frozen positions in the rate-profile. This approach enhances performance for medium-length codes, with an overall improvement of 0.12 dB.

Paper Structure

This paper contains 12 sections, 1 theorem, 14 equations, 5 figures, 1 table.

Table of Contents

  1. INTRODUCTION
  2. PRELIMINARIES
  3. Polar codes, CRC-Polar Codes, and PAC Codes
  4. Minimum Weight Codewords in Cosets
  5. Error Occurrence in List Decoding
  6. Reserving Large-index Coordinates in Rate-profile to Enhance Weight Distribution
  7. Reduction of the number of MWCs in CRC-polar Codes
  8. Proposed Schemes
  9. Profile-Shifted PAC Codes
  10. Continuous CRC-Polar Codes
  11. Numerical Results
  12. CONCLUSION

Key Result

Lemma 1

(rowshan2023minimum) For any coset $\mathcal{C}_i(\mathcal{I})$ where we have In other words, any cosets $\mathcal{C}_i$ where there is no frozen row $\mathbf{g}_f$ for $f \in \mathcal{I}^c \cap(i, N-1]$ such that $|\mathop{\mathrm{supp}}\nolimits(\mathop{\mathrm{bin}}\nolimits(f)) \backslash \mathop{\mathrm{supp}}\nolimits(\mathop{\mathrm{bin}}\nolimits(i))|>1$, we get $A_{i

Figures (5)

  • Figure 1: Precoding and encoding of polar codes.
  • Figure 2: Constructions of polar and PS-PAC codes in reliability order.
  • Figure 3: Construction of CCRC-polar codes in natural order.
  • Figure 4: Performance of (64, 32) and (64, 48) codes.
  • Figure 5: Performance of codes with $N = 256, \, 512$.

Theorems & Definitions (4)

  • Definition 1
  • Lemma 1
  • Remark 1
  • Example 1