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PAC Codes: Sequential Decoding vs List Decoding

Mohammad Rowshan, Andreas Burg, Emanuele Viterbo

TL;DR

This work addresses improving finite-length error correction and decoding efficiency for polar codes by introducing polarization-adjusted convolutional (PAC) codes. It advances PAC decoding with both list and Fano strategies, incorporating an adaptive path metric and constrained tree-search to reduce average decoding time while incurring modest FER degradation. Key contributions include partial rewind for SC backtracking, an adaptive bias in the Fano metric, and multiple backtracking/search constraints that yield substantial complexity reductions (roughly 50–80% on average). Distance-spectrum analysis and numerical results show PAC codes can significantly improve distance properties and approach dispersion bounds under Fano decoding, often outperforming standard polar codes at similar lengths. The findings provide practical guidance for implementing PAC codes and highlight the tradeoffs between pre-transform design, rate-profiles, and decoding architecture.

Abstract

In the Shannon lecture at the 2019 International Symposium on Information Theory (ISIT), Arıkan proposed to employ a one-to-one convolutional transform as a pre-coding step before the polar transform. The resulting codes of this concatenation are called polarization-adjusted convolutional (PAC) codes. In this scheme, a pair of polar mapper and demapper as pre- and postprocessing devices are deployed around a memoryless channel, which provides polarized information to an outer decoder leading to improved error correction performance of the outer code. In this paper, the list decoding and sequential decoding (including Fano decoding and stack decoding) are first adapted for use to decode PAC codes. Then, to reduce the complexity of sequential decoding of PAC/polar codes, we propose (i) an adaptive heuristic metric, (ii) tree search constraints for backtracking to avoid exploration of unlikely sub-paths, and (iii) tree search strategies consistent with the pattern of error occurrence in polar codes. These contribute to the reduction of the average decoding time complexity from 50% to 80%, trading with 0.05 to 0.3 dB degradation in error correction performance within FER=10^-3 range, respectively, relative to not applying the corresponding search strategies. Additionally, as an important ingredient in Fano decoding of PAC/polar codes, an efficient computation method for the intermediate LLRs and partial sums is provided. This method is effective in backtracking and avoids storing the intermediate information or restarting the decoding process. Eventually, all three decoding algorithms are compared in terms of performance, complexity, and resource requirements.

PAC Codes: Sequential Decoding vs List Decoding

TL;DR

This work addresses improving finite-length error correction and decoding efficiency for polar codes by introducing polarization-adjusted convolutional (PAC) codes. It advances PAC decoding with both list and Fano strategies, incorporating an adaptive path metric and constrained tree-search to reduce average decoding time while incurring modest FER degradation. Key contributions include partial rewind for SC backtracking, an adaptive bias in the Fano metric, and multiple backtracking/search constraints that yield substantial complexity reductions (roughly 50–80% on average). Distance-spectrum analysis and numerical results show PAC codes can significantly improve distance properties and approach dispersion bounds under Fano decoding, often outperforming standard polar codes at similar lengths. The findings provide practical guidance for implementing PAC codes and highlight the tradeoffs between pre-transform design, rate-profiles, and decoding architecture.

Abstract

In the Shannon lecture at the 2019 International Symposium on Information Theory (ISIT), Arıkan proposed to employ a one-to-one convolutional transform as a pre-coding step before the polar transform. The resulting codes of this concatenation are called polarization-adjusted convolutional (PAC) codes. In this scheme, a pair of polar mapper and demapper as pre- and postprocessing devices are deployed around a memoryless channel, which provides polarized information to an outer decoder leading to improved error correction performance of the outer code. In this paper, the list decoding and sequential decoding (including Fano decoding and stack decoding) are first adapted for use to decode PAC codes. Then, to reduce the complexity of sequential decoding of PAC/polar codes, we propose (i) an adaptive heuristic metric, (ii) tree search constraints for backtracking to avoid exploration of unlikely sub-paths, and (iii) tree search strategies consistent with the pattern of error occurrence in polar codes. These contribute to the reduction of the average decoding time complexity from 50% to 80%, trading with 0.05 to 0.3 dB degradation in error correction performance within FER=10^-3 range, respectively, relative to not applying the corresponding search strategies. Additionally, as an important ingredient in Fano decoding of PAC/polar codes, an efficient computation method for the intermediate LLRs and partial sums is provided. This method is effective in backtracking and avoids storing the intermediate information or restarting the decoding process. Eventually, all three decoding algorithms are compared in terms of performance, complexity, and resource requirements.

Paper Structure

This paper contains 25 sections, 21 equations, 12 figures, 1 table, 6 algorithms.

Table of Contents

  1. INTRODUCTION
  2. PRELIMINARIES
  3. Polar Codes and List Decoding
  4. Convolutional Codes and Fano Decoding
  5. Polarization-adjusted Codes
  6. PAC Encoding
  7. PAC List Decoding
  8. Fano Decoding of PAC Codes
  9. Partial Rewind of SC Algorithm
  10. Heuristic Path Metric
  11. Low-complexity Fano Decoding
  12. Adaptive Path Metric
  13. Constrained Tree Search
  14. Constraint on Number of Diversions from Best Path
  15. Exploring a Subset of bad branches
  16. ...and 10 more sections

Figures (12)

  • Figure 1: Code Concatenation
  • Figure 2: Successive cancellation factor graph for $N=4$
  • Figure 3: PAC coding scheme
  • Figure 4: Decoding tree: $\mu_j$s are the path metrics of the current best path (solid thick line) from the root to a node at level $j$ and the $\mu^\prime_j$s are the path metrics of the branches (solid thin line) diverging from the current best path.
  • Figure 5: Bottom-up backtracking
  • ...and 7 more figures