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IEEE Communications Letters

On Tail-Biting Polarization-Adjusted Convolutional (TB-PAC) Codes and Small-Sizes List Decoding

Bowen Feng, Yi Yang, Jian Jiao, Qinyu Zhang

Published Nov 28, 2022• 5 citations
Publisher

Abstract

Polarization-adjusted convolutional (PAC) coding is a good concatenation of convolutional codes and polar codes. For short block-length codes ($N\leq 128$), PAC codes with a sequential decoding or a list decoding with large list-sizes ($L\geq 128$) 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 ($L\leq 8$). 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.

DOI
10.1109/lcomm.2022.3225177

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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.
Polar codes have attracted the attention of numerous researchers in the past decade due to their excellent performance. However, their performance at short block lengths under standard successive cancellation decoding is far from desirable. An effective method to improve the performance at short lengths is CRC precoding followed by successive-cancellation list decoding. Later, Arikan presented polarization-adjusted convolutional (PAC) codes, which further improve the performance of polar codes. In fact, bit-flipping is another post-processing method that can improve decoding performance. In this paper, we propose a novel SCL Bit-Flipping of PAC Codes. We show that better performance can be achieved using list decoding when the list size is the same for PAC codes (N=128, K=64). The decoding performance of our newly proposed PAC-SCLF with a list size of 32 is 0.3 dB better than that of the traditional PAC-SCL with a list size of 32. We set the maximum number of bit flips to 5. The performance of the list size (L=32) for PAC-SCLF is almost the same as the performance of the list size (L=128) for PAC-SCL.
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.
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.
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 have been proposed to significantly improve error correction ability compared to polar codes by incorporating rate-1 convolutional precoding. However, the development of computationally efficient decoders for PAC codes is still an ongoing research area. Ordered Reliability Bit Guessing Random Additive Noise Decoding (ORB-GRAND), a universal near Maximum Likelihood (near-ML) decoder, has been introduced for decoding moderately redundant block codes by querying a list of putative test error patterns (TEP) in a fixed order, i.e. logistic weight order (LWO) and improved logistic weight order (iLWO). In this study, we undertake a comprehensive investigation into the performance of ORB-GRAND concerning high-rate PAC codes of short-to-medium length. Firstly, we conduct a comparative analysis between ORB-GRAND and successive cancellation list (SCL) decoding for PAC codes and integrate the Cyclic Redundancy Check (CRC) mechanism into ORB-GRAND. Distance properties of PAC and CRC-Aided PAC (CA-PAC) codes are explored to demonstrate the performance results. Secondly, we propose a novel TEP scheduling scheme in response to the distinctive reduction in BLER revealed by scheduling TEP in iLWO, particularly for short PAC codes under low Signal-to-Noise Ratio (SNR) conditions. This remarkable scheme demonstrates its efficacy by effectively mitigating the aforementioned BLER degradation while maintaining the exceptional performance of iLWO in high SNR conditions.
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.