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IEICE Transactions on Fundamentals of Electronics Communications and Computer Sciences

Parity-Check Polarization-Adjusted Convolutional Coding

Qingping Yu, You ZHANG, Renze Luo, Longye WANG, Xingwang Li

Published Jul 26, 2023• 2 citations
Publisher

Abstract

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.

DOI
10.1587/transfun.2023eal2036

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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.
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.
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.
The Polar-Adjusted Convolutional (PAC) codes, improve the error correction ability of the polar code by concatenating convolutional transformation and polar transformation. For the traditional stack decoding algorithm, the ideal performance has not yet been achieved for some PAC codes. In this paper, we adopt a multi-stack decoding algorithm to further improve the error performance of PAC codes. In the multi-stack decoding algorithm, the original single stack is converted into multiple stacks, which expands the exploration path of decoding, obtains more possible decoding results, and improves the decoding performance of PAC codes. Compared to the single-stack decoding, the multi-stack decoding achieves a performance gain of approximately 0.2dB for a PAC code with code length 128 and rate 0.5.
Polarization-adjusted convolutional (PAC) coding is a good concatenation of convolutional codes and polar codes. For short block-length codes (N≤128N\leq 128N≤128), PAC codes with a sequential decoding or a list decoding with large list-sizes (L≥128L\geq 128L≥128) can achieve better block-error-rate (BLER) performance than polar codes with CRC-aided list decoding. However, the PAC codes have no BLER performance advantage when the list decoding is with small list-sizes (L≤8L\leq 8L≤8). In this letter, tail-biting PAC (TB-PAC) coding is proposed, in which tail-biting convolutional pre-transformation is applied. A multi-round list decoding with small list-sizes is provided for TB-PAC codes, in which an early-stopping method is designed to control the number of rounds. In each round, the tail-biting bits are assigned various initial path-values and also used as a check at the end, which helps the correct path be distinguishable. The proposed small-sizes list decoding can help the TB-PAC codes achieve much better BLER performance than the polar/PAC codes.
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.