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arXiv:2202.12885 · IEEE International Symposium on Information Theory (ISIT)

Approximate Weight Distribution of Polarization-Adjusted Convolutional (PAC) Codes

Sadra Seyedmasoumian, Tolga M. Duman

Published Jun 26, 2022• 15 citations
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Abstract

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.

DOI
10.1109/isit50566.2022.9834587

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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.
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 have better error-correcting performance than polar codes mostly because of the improved weight distribution brought by the convolutional pre-transformation. In this paper, we propose the parity check PAC (PC-PAC) codes to further improve error-correcting performance of PAC codes. The design principle is to establish parity check functions between bits with distinct row weights, such that information bits of lower reliability are re-protected by the PC relation. Moreover, an algorithm to select which bits to be involved in parity-check functions is also proposed to make sure that the constructed codes have fewer minimum-weight codewords. Simulation results show that the proposed PC-PAC codes can achieve nearly 0.2dB gain over PAC codes at frame error rate (FER) about 10-3 codes.
While sequential decoding of polarization-adjusted convolutional (PAC) codes constructed using polar rate profile has low computational complexity, their error-correction per-formance falls far short of the theoretical bounds for finite blocklength codes. In this paper, we use the bit-flipping technique in the stack decoding algorithm of PAC codes in order to improve their error-correction performance. This technique maintains the low memory requirements of stack decoding and polar demapper. Additionally, at high SNR values, the number of visits at each level of the decoding tree is almost one. Numerical findings indicate that this approach is capable of outperforming the stack decoding algorithm.
In polarization-adjusted convolutional (PAC) codes, a convolutional transform is employed as a precoding stage before the polar transform. It turned out that the precoding stage improves the weight distribution of polar codes according. That could be the main reason for the superiority of error correction performance of PAC codes relative to polar codes. In this work, we explicitly show why the convolutional precoding reduces the number of minimum-weight codewords. Furthermore, we show where the precoding stage is not effective in the improvement of weight distribution. Eventually, to evaluate the FER gain obtainable from reducing the number of min-weight codewords, we assess the performance and the number of minimum-weight codewords resulted from an irregular convolutional precoding process.
Despite the extreme error-correction performance, the amount of computation of sequential decoding of the polarization-adjusted convolutional (PAC) codes is random. In sequential decoding of convolutional codes, the cutoff rate denotes the region between rates whose average computational complexity of decoding is finite and those which is infinite. In this paper, by benefiting from the polarization and guessing techniques, we prove that the required computation in sequential decoding of pre-transformed polar codes polarizes, and this polarization determines which set of bit positions within the rate profile may result in high computational complexity. Based on this, we propose a technique for taming the Reed-Muller (RM) rate-profile construction, and the performance results demonstrate that the error-correction performance of the PAC codes can achieve the theoretical bounds using the tamed-RM rate-profile construction and requires a significantly lower computational complexity than the RM rate-profile construction.
In this letter, we introduce an efficient method for estimating weight distributions of polar codes and polarization-adjusted convolutional (PAC) codes. Based on a recursive algorithm of computing the weight enumerating functions of polar cosets, this method focuses on two key objectives: accurately determining the number of low-weight codewords and quickly approximating the distribution of high-weight codewords. Simulation results demonstrate that this hybrid method maintains competitively low complexity while effectively achieving the objectives.