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Efficient Low-Memory Fast Stack Decoding with Variance Polarization for PAC Codes

Mohsen Moradi, Hessam Mahdavifar

Abstract

Polarization-adjusted convolutional (PAC) codes have recently emerged as a promising class of error-correcting codes, achieving near-capacity performance particularly in the short block-length regime. In this paper, we propose an enhanced stack decoding algorithm for PAC codes that significantly improves parallelization by exploiting specialized bit nodes, such as rate-0 and rate-1 nodes. For a rate-1 node with $N_0$ leaf nodes in its corresponding subtree, conventional stack decoding must either explore all $2^{N_0}$ paths, or, same as in fast list decoding, restrict attention to a constant number of candidate paths. In contrast, our approach introduces a pruning technique that removes candidate paths with small path metrics while ensuring that the probability of pruning the correct path decays exponentially with the threshold. Furthermore, we propose a novel approximation method for estimating variance polarization under the binary-input additive white Gaussian noise (BI-AWGN) channel. Leveraging these approximations, we develop an efficient stack-pruning strategy that selectively preserves decoding paths whose bit-metric values align with their expected means. This targeted pruning substantially reduces the number of active paths in the stack, thereby decreasing both decoding latency and computational complexity. Numerical results demonstrate that for a PAC$(128,64)$ code, our method achieves up to a 70\% reduction in the average number of paths without degrading error-correction performance.

Efficient Low-Memory Fast Stack Decoding with Variance Polarization for PAC Codes

Abstract

Polarization-adjusted convolutional (PAC) codes have recently emerged as a promising class of error-correcting codes, achieving near-capacity performance particularly in the short block-length regime. In this paper, we propose an enhanced stack decoding algorithm for PAC codes that significantly improves parallelization by exploiting specialized bit nodes, such as rate-0 and rate-1 nodes. For a rate-1 node with leaf nodes in its corresponding subtree, conventional stack decoding must either explore all paths, or, same as in fast list decoding, restrict attention to a constant number of candidate paths. In contrast, our approach introduces a pruning technique that removes candidate paths with small path metrics while ensuring that the probability of pruning the correct path decays exponentially with the threshold. Furthermore, we propose a novel approximation method for estimating variance polarization under the binary-input additive white Gaussian noise (BI-AWGN) channel. Leveraging these approximations, we develop an efficient stack-pruning strategy that selectively preserves decoding paths whose bit-metric values align with their expected means. This targeted pruning substantially reduces the number of active paths in the stack, thereby decreasing both decoding latency and computational complexity. Numerical results demonstrate that for a PAC code, our method achieves up to a 70\% reduction in the average number of paths without degrading error-correction performance.

Paper Structure

This paper contains 15 sections, 3 theorems, 18 equations, 4 figures, 3 tables, 8 algorithms.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Notation Convention
  4. PAC Coding Scheme
  5. Rate Profile
  6. Channel Parameters
  7. Stack Decoding
  8. Path Metric
  9. BI-AWGN
  10. Sequential Decoders for PAC Codes
  11. Polarization of Channel Parameters
  12. Stack pruning method
  13. Fast stack algorithm
  14. Numerical Results
  15. Conclusion

Key Result

Theorem 1

moradi2024fast. In a BI-DMC, the bit-metric random variable $\gamma(U_i; \mathbf{Y}, \mathbf{U}^{i-1})$ converges to its expected value in probability. More precisely, it holds that where $m_i$ is a positive real number.

Figures (4)

  • Figure 1: FER performance and the average number of paths in the stack comparison of PAC$(128,64)$ codes with different path-pruning strategies for the stack decoding algorithm.
  • Figure 2: FER performance comparison of PAC$(64,57)$ and RM$(64,57)$ codes under different decoding algorithms.
  • Figure 3: FER performance comparison of PAC$(128,99)$ codes with Fano and stack decoding algorithms.
  • Figure 4: FER performance comparison of PAC$(64,32)$ codes with Fano and stack decoding algorithms.

Theorems & Definitions (3)

  • Theorem 1
  • Theorem 2
  • Theorem 3