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arXiv:2208.04010 · IEEE Transactions on Communications

Application of Guessing to Sequential Decoding of Polarization-Adjusted Convolutional (PAC) Codes

Mohsen Moradi

Published May 29, 2023• 11 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1109/tcomm.2023.3280548

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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.
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.
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.
We propose an approach to understand and exploit Polarization-Adjusted Convolutional (PAC) Codes that is directly tied to decoders with memory, specifically Successive Cancellation List (SCL) decoding. The crux of our approach is to use a modified Density Evolution Gaussian Approximation (DEGA) to account for errors in the decoding process and accurately track the path metrics (PMs) likely to incur decoding errors. Our approach not only explains the benefits provided by the use of the rate one precoding, but also provides new insight into why certain information sets perform well under PAC. We leverage the approach to design new information sets, and in particular, we design a$(128,42,128,42,128,42,L=8)codethatoffershalfacode that offers half acodethatoffershalfa\text{dB}gainoverthestate−of−the−artataFrameErrorRate(FER)ofgain over the state-of-the-art at a Frame Error Rate (FER) ofgainoverthestate−of−the−artataFrameErrorRate(FER)of\overline{1}0^{-3}andoutperformstheReed−Muller(RM)setwithand outperforms the Reed-Muller (RM) set withandoutperformstheReed−Muller(RM)setwithL=32$. We also draw connections between our approach and works studying the minimum weight of PAC codes.
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.