[Paper Review] A Monte-Carlo Based Construction of Polarization-Adjusted Convolutional (PAC) Codes

Mohsen Moradi, Amir Mozammel|arXiv (Cornell University)|Jun 15, 2021

Advanced Wireless Communication TechniquesEngineering17 citations

TL;DR

This paper proposes a Monte-Carlo-based rate-profile construction method for Polarization-Adjusted Convolutional (PAC) codes that improves error-correction performance while guaranteeing low mean sequential decoding complexity beyond a target SNR. By selecting information bits based on channel cutoff rates and iterative refinement, the method achieves up to 0.5 dB coding gain over RM-Polar and polar rate profiles at FER = 10⁻³.

ABSTRACT

This paper proposes a rate-profile construction method for polarization-adjusted convolutional (PAC) codes of any code length and rate, which is capable of maintaining trade-off between the error-correction performance and decoding complexity of PAC code. The proposed method can improve the error-correction performance of PAC codes while guaranteeing a low mean sequential decoding complexity for signal-to-noise ratio (SNR) values beyond a target SNR value.

Motivation & Objective

• To address the trade-off between error-correction performance and decoding complexity in PAC codes.
• To develop a rate-profile construction method that maintains low mean sequential decoding complexity for SNR values above a target.
• To improve PAC code performance beyond that of conventional polar and Reed-Muller rate profiles.
• To enable practical implementation of PAC codes via guaranteed finite mean complexity under sequential decoding.
• To generalize the method to any pre-transformed polar code by replacing the convolutional encoder matrix.

Proposed method

• The method uses a Monte-Carlo simulation-based approach to iteratively refine the rate profile by evaluating candidate indices based on their Bhattacharyya parameters and cutoff rates.
• It starts with an initial rate profile derived from the channel cutoff rate at a specified construction SNR, ensuring low complexity for SNR values above that point.
• At each iteration, the algorithm removes indices with high Bhattacharyya parameters and selects new candidates to improve performance while maintaining cutoff-rate compliance.
• The 1-bit quantization function q(x, δ) is used to model the reliability of bit-channels, with δ set to 0.5 to define the cutoff-rate threshold.
• The final rate profile is selected based on the best error-correction performance at the target SNR, while ensuring it lies above the cutoff-rate profile at that SNR.
• The method is generalizable and can be applied to any PAC code by replacing the convolutional encoder matrix with a desired transformation matrix.

Figure 1: Block diagram of PAC coding scheme.

Experimental results

Research questions

• RQ1 Can a rate-profile construction method for PAC codes simultaneously improve error-correction performance and maintain low sequential decoding complexity?
• RQ2 How can the cutoff rate phenomenon be leveraged to ensure finite mean decoding complexity beyond a target SNR?
• RQ3 What is the performance gain of the proposed method compared to RM-Polar and polar rate profiles in terms of FER and complexity?
• RQ4 How does increasing the construction SNR affect the final rate profile and its performance-complexity trade-off?
• RQ5 Can the proposed method be generalized to other pre-transformed polar codes by modifying the generator matrix?

Key findings

• The proposed Monte-Carlo-based rate-profile construction method achieves a 0.5 dB coding gain at FER = 10⁻³ for both PAC(256,128) and PAC(64,32) codes compared to RM-Polar and polar rate profiles.
• For PAC(256,128), the MC-3dB profile achieves a 0.5 dB gain over RM-Polar at FER = 10⁻³ while maintaining low mean sequential decoding complexity beyond 3 dB SNR.
• The MC-5dB profile for PAC(64,32) achieves FER performance closer to the RCU bound but with increased decoding complexity, as it lies below the cutoff-rate profile at 3 dB.
• The ANV (average number of visited nodes) remains low beyond the target SNR for all proposed profiles, confirming finite mean complexity under Fano decoding.
• The method successfully shifts the low-complexity region to higher SNR values when the construction SNR is increased, while preserving performance gains.
• The method is general and can be applied to conventional polar codes by replacing the convolutional encoder with an identity matrix.

Figure 2: Error frequency of subsequent bits of FBE for PAC $(256,128)$ code.