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Simplified Successive Cancellation List Decoding of PAC Codes

Hamid Saber, Homayoon Hatami, Jung Hyun Bae

TL;DR

This work introduces a simplified SCL decoding framework for polarization-adjusted convolutional (PAC) codes by leveraging special-node processing that accounts for the CC encoder. The SSCL PAC decoder reduces decoding latency by performing CC-aware operations at carefully chosen nodes (Rate-0, Repetition, Rate-1, SPC) and employs an inverse CC encoding step with a nested, upper-triangular Toeplitz structure to generate list candidates efficiently. Empirical results show that PAC codes under SSCL achieve nearly the same BLER as SCL while reducing latency by up to about 41%, offering practical gains for low-latency communications. The approach extends the well-known SSCL concepts from polar codes to PAC codes, enabling faster decoding without sacrificing performance and with manageable complexity thanks to the nested inverse CC property and shared hardware for the inverse transformer.

Abstract

Polar codes are the first class of structured channel codes that achieve the symmetric capacity of binary channels with efficient encoding and decoding. In 2019, Arikan proposed a new polar coding scheme referred to as polarization-adjusted convolutional (PAC)} codes. In contrast to polar codes, PAC codes precode the information word using a convolutional code prior to polar encoding. This results in material coding gain over polar code under Fano sequential decoding as well as successive cancellation list (SCL) decoding. Given the advantages of SCL decoding over Fano decoding in certain scenarios such as low-SNR regime or where a constraint on the worst case decoding latency exists, in this paper, we focus on SCL decoding and present a simplified SCL (SSCL) decoding algorithm for PAC codes. SSCL decoding of PAC codes reduces the decoding latency by identifying special nodes in the decoding tree and processing them at the intermediate stages of the graph. Our simulation results show that the performance of PAC codes under SSCL decoding is almost similar to the SCL decoding while having lower decoding latency.

Simplified Successive Cancellation List Decoding of PAC Codes

TL;DR

This work introduces a simplified SCL decoding framework for polarization-adjusted convolutional (PAC) codes by leveraging special-node processing that accounts for the CC encoder. The SSCL PAC decoder reduces decoding latency by performing CC-aware operations at carefully chosen nodes (Rate-0, Repetition, Rate-1, SPC) and employs an inverse CC encoding step with a nested, upper-triangular Toeplitz structure to generate list candidates efficiently. Empirical results show that PAC codes under SSCL achieve nearly the same BLER as SCL while reducing latency by up to about 41%, offering practical gains for low-latency communications. The approach extends the well-known SSCL concepts from polar codes to PAC codes, enabling faster decoding without sacrificing performance and with manageable complexity thanks to the nested inverse CC property and shared hardware for the inverse transformer.

Abstract

Polar codes are the first class of structured channel codes that achieve the symmetric capacity of binary channels with efficient encoding and decoding. In 2019, Arikan proposed a new polar coding scheme referred to as polarization-adjusted convolutional (PAC)} codes. In contrast to polar codes, PAC codes precode the information word using a convolutional code prior to polar encoding. This results in material coding gain over polar code under Fano sequential decoding as well as successive cancellation list (SCL) decoding. Given the advantages of SCL decoding over Fano decoding in certain scenarios such as low-SNR regime or where a constraint on the worst case decoding latency exists, in this paper, we focus on SCL decoding and present a simplified SCL (SSCL) decoding algorithm for PAC codes. SSCL decoding of PAC codes reduces the decoding latency by identifying special nodes in the decoding tree and processing them at the intermediate stages of the graph. Our simulation results show that the performance of PAC codes under SSCL decoding is almost similar to the SCL decoding while having lower decoding latency.
Paper Structure (16 sections, 1 theorem, 26 equations, 3 figures, 4 tables, 3 algorithms)

This paper contains 16 sections, 1 theorem, 26 equations, 3 figures, 4 tables, 3 algorithms.

Table of Contents

  1. Introduction
  2. Background
  3. Arikan's PAC Codes
  4. SCL decoding of PAC codes
  5. Simplified Successive Cancellation List Decoding of PAC Codes
  6. Recursive SC decoding on the decoding tree
  7. Rate-0 node
  8. Repetition node
  9. Rate-1 node
  10. Single Parity Check (SPC) node
  11. Numerical Results
  12. Performance
  13. Complexity
  14. Conclusion
  15. Appendix
  16. ...and 1 more sections

Key Result

Lemma 3.1

For any upper-triangular Toeplitz matrix $\mathbf{G}_{cc,n_\nu}$, the inverse matrix $\mathbf{G}_{cc,n_\nu}^{-1}$ is an upper-triangular Toeplitz matrix and takes the following form fro some $N_\nu \times N_\nu$ matrix $\mathbf{P}_{n_\nu}$.

Figures (3)

  • Figure 1: Decoding tree of a length-4 PAC code
  • Figure 2: BLER performance of (128,72) PAC code and CRC-aided polar code under SCL decoding.
  • Figure 3: BLER performance of (256,128) PAC code and CRC-aided polar code under SCL decoding.

Theorems & Definitions (1)

  • Lemma 3.1