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arXiv:2307.05871

A Novel SCL Bit-Flipping Decoding Of Polarization-Adjusted Convolutional (PAC) Codes

Zhang, Wei

Published Jul 12, 2023• 1 citation
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Abstract

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.

DOI
10.48550/arxiv.2307.05871

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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.
By adopting successive cancellation list decoding (SCL), polar codes demonstrate competitive error correction performance over LDPC and Turbo codes. However, SCL decoding suffers from high computational complexity and long decoding latency, especially when the list size is very large. Successive cancellation flip (SCF), as another decoding algorithm that can achieve high error correction performance, has a complexity that is close to that of successive cancellation (SC) decoding. With the observation that SCL and SCF decoding are similar at giving more chances to inspect possible codewords simultaneously or sequentially, a novel hybrid decoder is proposed in this paper, which essentially combines the ideas of SCF and SCL decoders. Moreover, in order to compensate for the degradation of performance caused by the reduction of path splitting and further reduce the decoding latency, the convolutional polar codes are adopted with a designed bit-flipping set. Simulation results demonstrate that the proposed decoder achieves the reduction of decoding latency while attaining better performance than conventional CRC-aided SCL decoder.
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, as the coding scheme for the control channel in fifth-generation mobile communication technology (5G), have attracted widespread attention since their proposal. As a mainstream decoding algorithm for polar codes, the successive cancellation list (SCL) decoder usually improves the error correction performance by increasing the list size, but this method suffers from the problems of high decoding complexity. To address this problem, this paper proposes a layered-search bit-flipping (LS-SCLF) decoding algorithm based on SCL decoding. Firstly, a new flip-bit metric is proposed, which derives a formula to approximate the probability of an error occurring in an information bit. This formula introduces a perturbation parameter to improve the calculation accuracy. Secondly, a compromise scheme for determining the perturbation parameter is proposed. The scheme uses Monte Carlo simulation to determine an optimized parameter for the precise positioning of the first erroneous decoded bit under different decoding conditions. Finally, a layered search strategy is adopted to sequentially search the erroneous decoded bits from the low order to high order, which can correct up to multiple bits at the same time. Simulation results show that the proposed algorithm achieves improved error correction performance with a slight increase in decoding complexity compared to the generalized SCL-Flip (GSCLF) decoding algorithm. This algorithm also achieves a good balance between the error correction performance and decoding complexity.
The eBCH-polar codes is a powerful way to construct polar codes. With high minimum Hamming distance, it outperforms ordinary Polar codes under successive cancellation list (SCL) decoder. However, the absence of cyclic redundancy check (CRC) bits in eBCH-polar codes makes it challenging to adopt SCL-flip (SCLF) decoding scheme. In this research, we propose a flip scheme activated by a Normalized Path Metric (PM) range and a mismatch criterion, which respectively leverage the properties of SCL decoding and dynamic frozen bits of eBCH-polar codes. This work also improves the selection the flipping bits by modifying metric to account for the structure of dynamic frozen bits and reduce the search space. Based on these modifications, we proposed SCLF-1 and SCLF-2 decoding algorithms for eBCH-polar codes. Simulation results show that both of the proposed decoding schemes achieve superior performance compared to other contemporary decoding algorithms for CRC aided polar (CA-polar) codes, demonstrating enhanced error rates performance within the same decoding attempt.
A successive cancellation list (SCL) decoder with the assistance of a cyclic redundancy check (CRC) can provide competitive decoding performance for polar codes compared with the state-of-the-art codes. In this paper, we adopt a bit-flip algorithm for the CRC-assisted SCL decoder to further improve the error-correcting performance. We first introduce a new bit-flip criterion for the SCL decoder. To support this criterion, we give the metric to identify the positions that need to be flipped. To verify the superiority of the proposed bit-flip criterion, we also consider a kind of SCL-flip decoder whose criterion is very similar with the existing one and we will prove that its performance is worse than the decoder with the proposed bit-flip criterion. The simulation results show that the decoder with the proposed bit-flip criterion outperforms both the standard CRC-assisted SCL decoder and the existing SCL-based bit-flip decoder.
This work presents a fast successive-cancellation list flip (Fast-SCLF) decoding algorithm for polar codes that addresses the high latency issue associated with the successive-cancellation list flip (SCLF) decoding algorithm. We first propose a bit-flipping strategy tailored to the state-of-the-art fast successive-cancellation list (FSCL) decoding that avoids tree-traversal in the binary tree representation of SCLF, thus reducing the latency of the decoding process. We then derive a parameterized path selection error model to accurately estimate the bit index at which the correct decoding path is eliminated from the initial FSCL decoding. The trainable parameter is optimized online based on an efficient supervised learning framework. Simulation results show that for a polar code of length 512 with 256 information bits, with similar error-correction performance and memory consumption, the proposed Fast-SCLF decoder reduces up to 73.4% of the average decoding latency of the SCLF decoder with the same list size at the frame error rate of 10−4, while incurring a maximum computational complexity overhead of 27.6%. For the same polar code of length 512 with 256 information bits and at practical signal-to-noise ratios, the proposed decoder with list size 4 reduces 89.3% and 43.7% of the average complexity and decoding latency of the FSCL decoder with list size 32 (FSCL-32), respectively, while also reducing 83.2% of the memory consumption of FSCL-32. The significant improvements of the proposed decoder come at the cost of 0.07 dB error-correction performance degradation compared with FSCL-32.