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IEEE Transactions on Communications

A Path Probability Perspective on Rate-Profiling Design for Polarization-Adjusted Convolutional (PAC) Codes Under List Decoding

Aolin Liu, Bowen Feng, Chulong Liang, Jin Xu, Ye Wang, Qinyu Zhang

Published Dec 30, 2024• 2 citations
Publisher

Abstract

A novel rate-profiling design is proposed for polarization-adjusted convolutional (PAC) codes under list decoding, adopting a path probability perspective. Inspired by the concepts of Hamming distance and Hamming weight, the concepts of path probability distance (PPD) and path probability weight (PPW) are innovatively introduced to measure the disparity between the erroneous paths and the correct path for PAC codes under probabilistic decoding. An in-depth analysis of the recursion of node log-likelihood ratios (LLRs) in the factor graph is conducted, elucidating the derivation of a lower bound on the PPW as a consequential outcome. Utilizing the concept of the PPW, a rate-profiling design algorithm is proposed to establish the reliability ranking for the bit-channels of PAC codes. Simulation results demonstrate that PAC codes employing the proposed rate-profiling exhibit advantages over alternative rate-profiling and other state-of-the-art polar coding schemes across various block lengths and code rates under short decoding list sizes.

DOI
10.1109/tcomm.2024.3523957

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IEEE Communications Letters•Aug 21, 2023•11 citations
A novel rate-profiling design is proposed for polarization-adjusted convolutional (PAC) codes under list decoding, adopting a path probability perspective. Inspired by the concepts of Hamming distance and Hamming weight, the concepts of path probability distance (PPD) and path probability weight (PPW) are innovatively introduced to measure the disparity between the erroneous paths and the correct path for PAC codes under probabilistic decoding. An in-depth analysis of the recursion of node log-likelihood ratios (LLRs) in the factor graph is conducted, elucidating the derivation of a lower bound on the PPW as a consequential outcome. Utilizing the concept of the PPW, a rate-profiling design algorithm is proposed to establish the reliability ranking for the bit-channels of PAC codes. Simulation results demonstrate that PAC codes employing the proposed rate-profiling exhibit advantages over alternative rate-profiling and other state-of-the-art polar coding schemes across various block lengths and code rates under short decoding list sizes.
Polarization-Adjusted Convolutional (PAC) codes achieve near-capacity performance at short block lengths, yet their error-correction capability depends strongly on where frozen bits are placed. The conventional Reed-Muller rule works well when a Successive Cancellation List (SCL) decoder can employ a large list, but it loses effectiveness when the list size must remain small. We introduce an alternative rate-profiling method that ranks bit positions by the average magnitude of their log-likelihood ratios collected in advance and designates the least reliable positions as frozen. Simulations of a PAC code with block length one hundred twenty-eight and rate one half over an additive white Gaussian noise channel show that the proposed profile yields lower bit- and frame-error rates than the Reed-Muller rule for list sizes up to sixteen, although its performance stops improving for larger lists. These results indicate that the LLR-based profile is a practical choice when decoder complexity limits the list size.
Polarization-adjusted convolutional (PAC) codes combine polar and convolutional transformations to enhance the distance properties of polar codes. They offer a performance very close to the finite-length information-theoretic bounds for short blocklengths. In this paper, we develop a method of computing the weight distribution of PAC codes in an approximate form by employing a probabilistic technique. We demonstrate that the results well match the exact weight distributions for small codes that can be computed using a brute-force algorithm. We also present a way of employing the results (along with a union bound on the code performance) to design specific PAC codes or, more precisely, to determine suitable rate profiles via simulated annealing. Numerical examples illustrate that the PAC codes with the designed rate profiles offer superior performance.
This paper introduces a novel rate-profile design for Polarization-Adjusted Convolutional (PAC) codes by simplifying the construction of an information set (inf set) for a PAC code of lengthNNNinto the design of inf sets forMMMcomponent codes of lengthNM\frac{N}{M}MN​. We define a Markov Decision Process (MDP) that formulates the task of identifying the inf set that minimizes the Block Error Rate (BLER). To achieve this, we employ a Deep QNetwork (DQN) with an experience replay buffer to determine the optimal inf set with the lowest BLER. Simulation results demonstrate that under Successive Cancellation List (SCL) decoding, our designed PAC codes improve BLER compared to state-of-the-art PAC codes. This method is not limited to the design of PAC code inf set; it is applicable to inf set design for the general family of precoded Polar codes and supports any decoder.
Polarization-adjusted convolutional (PAC) codes are special concatenated codes that utilize a one-to-one convolutional transform as a pre-coding step before the polar transform, thereby approaching the dispersion approximation for code (128, 64). However, current list decoders require an overly large list size to achieve near-bound performance, leading to unsatisfactory decoding latency. In this letter, we propose a fast list decoding algorithm with new special nodes for PAC codes, which effectively decreases decoding latency while preserving error-correction performance. Compared to the state-of-the-art work, the proposed algorithm can reduce decoding latency by up to 25.81% for the PAC code of length 128 with a code rate of 1/6.
In this paper, we propose a reinforcement learning based algorithm for rate-profile construction of Arikan’s Polarization Adjusted Convolutional (PAC) codes. This method can be used for any blocklength, rate, list size under successive cancellation list (SCL) decoding and convolutional precoding polynomial. To the best of our knowledge, we present, for the first time, a set of new reward and update strategies which help the reinforcement learning agent discover much better rate-profiles than those present in existing literature.Simulation results show that PAC codes constructed with the proposed algorithm perform better in terms of frame erasure rate (FER) compared to the PAC codes constructed with contemporary rate profiling designs for various list lengths. Further, by using a (64, 32) PAC code as an example, it is shown that the choice of convolutional precoding polynomial can have a significant impact on rate-profile construction of PAC codes.
A latest coding scheme named polarization-adjusted convolutional (PAC) codes is shown to provide dramatic improvement in performance as compared to polar codes under list decoding. However, to achieve this performance gain, the list size needs to be excessively large, which may cause insufferable latency. For PAC codes, utilizing special nodes can reduce the time complexity. In this letter, we propose an extension of the sequence repetition (SR) nodes to PAC codes to further reduce the decoding latency. First, we define the notion of convolution-effected repetition sequences, which helps us to decode a specific SR node. Then, we propose a fast list decoding algorithm by taking the SR nodes into consideration. Simulation results indicate that the performance of the proposed decoding algorithm is almost identical to the original list decoding algorithm while achieving a higher degree of parallelism. In particular, for a PAC code of length 128 with code rate 1/4, the decoding latency can be reduced by over 35% as compared to the state-of-the-art fast list decoder.