For polar codes, the statistical breakdown of belief propagation (BP) decoding errors is firstly proposed by Sunet al.The errors are classified into three categories: unconverged errors, false converged errors and oscillation errors, which are corrected by three different post-processing methods. As for the BP List (BPL) decoding, we discover that one type of error can be transformed to another type by different factor graphs in the failed BP decoding. Besides, false converged error is usually easier to be detected and modified. Therefore, we only target false converged error to detect and correct. In this work, we propose a two-level detection rule for false converged errors and its corresponding post-processing algorithm to modify these errors in order to improve the performance of the BPL decoding. Numerical results show that the error-correction performance of the proposed decoder is more than 0.85dB better than that of the permuted BPL (PBPL) decoder with slight extra computation complexity at the frame error rate (FER) of$10^{-5}$for 5G (1024,512) polar code.
For polar codes, the statistical breakdown of belief propagation (BP) decoding errors is firstly proposed by Sunet al.The errors are classified into three categories: unconverged errors, false converged errors and oscillation errors, which are corrected by three different post-processing methods. As for the BP List (BPL) decoding, we discover that one type of error can be transformed to another type by different factor graphs in the failed BP decoding. Besides, false converged error is usually easier to be detected and modified. Therefore, we only target false converged error to detect and correct. In this work, we propose a two-level detection rule for false converged errors and its corresponding post-processing algorithm to modify these errors in order to improve the performance of the BPL decoding. Numerical results show that the error-correction performance of the proposed decoder is more than 0.85dB better than that of the permuted BPL (PBPL) decoder with slight extra computation complexity at the frame error rate (FER) of10−5for 5G (1024,512) polar code.
In this paper, we present an improved belief propagation list (BPL) decoding algorithm for polar codes. Rather than getting L factor graphs (FGs) at random and cyclic shift permutation, we use the upper bounds on the block error propability of polar codes with different FGs as the metric to choose the best L FGs. By observing the bounds of different FGs, we propose a heuristic method to reduce search complexity. Simulation results show that there is only a gap of 0.2 dB between the frame error rate (FER) performance of the improved BPL decoder using RM16-GA construction and that of length-1024 5G polar code decoded by SCL with the same list size of 32 at FER =10-4. Moreover, with the proposed FG selection method, BPL decoding can reduce clock cycles by 97.74% compared with the SCL decoding.
Belief propagation (BP) decoding for polar codes has been extensively studied because of its inherent parallelism. However, its performance remains inferior to that of successive cancellation list decoding (SCL) due to the structure of the decoding graph. To improve the block error rate (BLER) performance, the BP correction (BPC) decoding, a post-processing scheme that corrects prior knowledge of the identified code bit, improves convergence by executing additional iterations on the failed BP decoder. Moreover, the BPC decoder demonstrates a better decoding performance than the BP-based bit-flipping decoder. Nevertheless, the additional decoding attempts lead to increased latency. In this article, a modified BPC decoder is proposed to reduce the number of decoding attempts by redefining the correction rules. A new metric is designed to effectively identify the corrected location. Numerical results show that the proposed modified BPC decoder achieves a slight improvement in BLER compared with the original BPC, with a dramatic reduction in average complexity. Furthermore, a higher-order version, named MBPC-Ω, is extended to further improve the performance, where the Ω is the maximum correction order. Numerical results show that the higher-order modified BPC achieves a similar BLER performance to existing multiple bit-flipping BP decoders but has around half the latency overhead. In addition, the proposed MBPC-2 decoder performs better than the cyclic redundancy check-aided SCL (CA-SCL) decoder with list size 4 and is slightly worse than the CA-SCL with list size 8 in high signal-to-noise ratio (SNR) regions but with significant decoding latency reduction.
We propose a belief propagation list (BPL) decoder with comparable performance to the successive cancellation list (SCL) decoder of polar codes, which already achieves the maximum likelihood (ML) bound of polar codes for sufficiently large list size L. The proposed decoder is composed of multiple parallel independent belief propagation (BP) decoders based on differently permuted polar code factor graphs. A list of possible transmitted codewords is generated and the one closest to the received vector, in terms of Euclidean distance, is picked. To the best of our knowledge, the proposed BPL decoder provides the best performance of plain polar codes under iterative decoding known so far. The proposed algorithm does not require any changes in the polar code structure itself, rendering the BPL into an alternative to the SCL decoder, equipped with a soft output capability enabling, e.g., iterative detection and decoding to further improve performance. Further benefits are the lower decoding latency than the SCL decoder and the possibility of high throughput implementations. Additionally, we show that a different selection strategy of frozen bit positions can further enhance the error-rate performance of the proposed decoder.
Although iterative decoding of polar codes has recently made huge progress based on the idea of permuted factor graphs, it still suffers from a non-negligible performance degradation when compared to state-of-the-art CRC-aided successive cancellation list (CA-SCL) decoding. In this work, we show that iterative decoding of polar codes based on the belief propagation list (BPL) algorithm can approach the error-rate performance of CA-SCL decoding and, thus, can be efficiently used for decoding the standardized 5G polar codes. Rather than only utilizing the cyclic redundancy check (CRC) as a stopping condition (i.e., for error-detection), we also aim to benefit from the error-correction capabilities of the outer CRC code. For this, we develop two distinct soft-decision CRC decoding algorithms: a Bahl-Cocke-Jelinek-Raviv (BCJR)-based approach and a sum product algorithm (SPA)-based approach. Further, an optimized selection of permuted factor graphs is analyzed and shown to reduce the decoding complexity significantly. Finally, we benchmark the proposed CRC-aided belief propagation list (CA-BPL) decoding to state-of-the-art 5G polar codes under CA-SCL decoding and, thereby, showcase an error-rate performance not just close to the CA-SCL but also close to the maximum likelihood (ML) bound as estimated by ordered statistic decoding (OSD).
To the best of our knowledge, Belief Propagation List (BPL) decoding achieves the best error-correction performance within the class of BP based polar code decoding algorithms. It regards different permutations of the polar code factor graph as a range of available "decoding lists" for individually executing the BP algorithm. Then, a minimum Euclidean distance criterion is employed to combine these individual decisions. However, we demonstrated that simply increasing the number of lists involved in the BPL algorithm does not always improve its performance. Instead, we exploit the extrinsic information transfer (EXIT) chart technology to verify the efficiency of a decoding list. Based on this analysis, an efficient list selection scheme is proposed. In the same spirit, in order to mitigate the impact of relatively inferior decoding lists on other selected decoding lists, new decision fusion schemes are also designed. Benefiting from these improvements of BPL algorithm, performance gains are evidenced in our simulation results.
Due to the design principle of parallel processing, belief propagation (BP) decoding is attractive, and it provides good error-correction performance compared with successive cancellation (SC) decoding. However, its error-correction performance is still inferior to that of successive cancellation list (SCL) decoding. Consequently, this paper proposes a novel flip-list- (FL)-enabled belief propagation (BP) method to improve the error-correction performance of BP decoding for polar codes with low computational complexity. The proposed technique identifies the vulnerable channel log-likelihood ratio (LLR) that deteriorates the BP decoding result. The FL is utilized to efficiently identify the erroneous channel LLRs and correct them for the next BP decoding attempt. The preprocessed channel LLR through FL improves the error-correction performance with minimal flipping attempts and reduces the computational complexity. The proposed technique was compared with the state-of-the-art BP, i.e., BP bit-flip (BP-BF), generalized BP-flip (GBPF), cyclic redundancy check (CRC)-aided (CA-SCL) decoding, and ordered statistic decoding (OSD), algorithms. Simulation results showed that the FL-BP had an excellent block error rate (BLER) performance gain up to 0.7 dB compared with BP, BP-BF, and GBPF decoder. Besides, the computational complexity was reduced considerably in the high signal-to-noise ratio (SNR) regime compared with the BP-BF and GBPF decoding methods.