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arXiv:2103.12483 · IEEE Globecom Workshops (GC Wkshps)

On Convolutional Precoding in PAC Codes

Mohammad Rowshan, Emanuele Viterbo

Published Dec 1, 2021• 26 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1109/gcwkshps52748.2021.9681987

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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.
Convolutional precoding in polarization-adjusted convolutional (PAC) codes is a recently introduced variant of polar codes. It has demonstrated an effective reduction in the number of minimum weight codewords (a.k.a error coefficient) of polar codes. This reduction has the potential to significantly improve the error correction performance. From a codeword formation perspective, this reduction has limitations. Capitalizing on the understanding of the decomposition of minimum-weight codewords, this paper studies reverse precoding that can effectively reduce minimum-weight codewords more than in PAC codes. We propose a look-ahead list decoding for the reverse PAC codes, which has the same order of complexity as list decoding in PAC codes. Through numerical analysis, we demonstrate a notable reduction in error coefficients compared to PAC codes and polar codes, resulting in a remarkable improvement in the block error rate, in particular at high code rates.
Convolutional precoding in polarization-adjusted convolutional (PAC) codes, a recently introduced variant of polar codes, has demonstrated an effective reduction in the number of minimum weight codewords (a.k.a error coefficient) of polar codes. This reduction has the potential to significantly improve the error correction performance. From a codeword formation perspective in cosets, this reduction has a limitation in the PAC coding which depends on the rows of the generator matrix involved in the formation of codewords. To overcome this limitation, capitalizing on the understanding of the decomposition of minimum-weight codewords, this paper introduces a novel precoding scheme that strategically disrupts the formation of a majority of minimum-weight codewords. This scheme significantly enhances the error coefficient without compromising the minimum distance of the code. Through numerical analysis, we demonstrate a noteworthy reduction in error coefficients compared to PAC codes and polar codes, resulting in a remarkable improvement in the block error rate of short codes.
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.
The number of minimum weight codewords, a.k.a error coefficient, is a good comparative measure for the block error rate (BLER) of a linear block code, in particular in a high SNR regime. The smaller the error coefficient, the lower the BLER. Unlike polar codes, the error coefficient of polarization-adjusted convolutional (PAC) codes cannot be determined easily due to the precoding stage. In this work, we propose an enumeration method that considers the impact of convolutional precoding on the minimum weight codewords of polar codes. This relative enumeration method simplifies the process significantly compared with conventional methods as the complexity analysis shows.
The minimum Hamming distance of a linear block code is the smallest number of bit changes required to transform one valid codeword into another. The code’s minimum distance determines the code’s error-correcting capabilities. Furthermore, The number of minimum weight codewords, a.k.a. error coefficient, gives a good comparative measure for the block error rate (BLER) of linear block codes with identical minimum distance, in particular at a high SNR regime under maximum likelihood (ML) decoding. A code with a smaller error coefficient would give a lower BLER. Unlike polar codes, a closed-form expression for the enumeration of the error coefficient of polarization-adjusted convolutional (PAC) codes is yet unknown. As PAC codes are convolutionally pre-transformed polar codes, we study the impact of pre-transformation on polar codes in terms of minimum Hamming distance and error coefficient by partitioning the codewords into cosets. We show that the minimum distance of PAC codes does not decrease; however, the pre-transformation may reduce the error coefficient depending on the choice of convolutional polynomial. We recognize the properties of the cosets where pre-transformation is ineffective in decreasing the error coefficient, giving a lower bound for the error coefficient. Then, we propose a low-complexity enumeration method that determines the number of minimum weight codewords of PAC codes relying on the error coefficient of polar codes. That is, given the error coefficient Awmin of polar codes, we determine the reduction X in the error coefficient due to convolutional pre-transformation in PAC coding and subtract it from the error coefficient of polar codes, Awmin - X. Furthermore, we numerically analyze the tightness of the lower bound and the impact of the choice of the convolutional polynomial on the error coefficient based on the sub-patterns in the polynomial’s coefficients. Eventually, we show how we can further reduce the error coefficient in the cosets.
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.