For the scenarios with high throughput requirements, the belief propagation (BP) decoding is one of the most promising decoding strategies for polar codes. By pruning the redundant variable nodes (VNs) and check nodes (CNs) in the original factor graph, the graph is condensed to a sparse bipartite graph which is similar to the graph for low-density parity-check (LDPC) codes. In this paper, we introduce the bit-flipping scheme into the LDPC-like BP (L-BP) decoding and propose two methods to identify the error-prone VNs. By additional decoding attempts, the L-BP flip (L-BPF) decoding improves the error-rate performance with a similar average complexity for high Eb=N0values. The simulation results show that the L-BPF decoding achieves 0:25 dB gain compared with the L-BP decoding.
For the scenarios with high throughput requirements, the belief propagation (BP) decoding is one of the most promising decoding strategies for polar codes. By pruning the redundant variable nodes (VNs) and check nodes (CNs) in the original factor graph, the graph is condensed to a sparse bipartite graph which is similar to the graph for low-density parity-check (LDPC) codes. In this paper, we introduce the bit-flipping scheme into the LDPC-like BP (L-BP) decoding and propose two methods to identify the error-prone VNs. By additional decoding attempts, the L-BP flip (L-BPF) decoding improves the error-rate performance with a similar average complexity for high Eb=N0values. The simulation results show that the L-BPF decoding achieves 0:25 dB gain compared with the L-BP decoding.
Belief propagation (BP) has attracted increasing attention due to its high parallelism. In this letter, a correction method is proposed to improve the block error rate (BLER) performance of the BP decoder for polar codes so that it is comparable to that of the successive cancellation list (SCL) decoder. Compared with BP flipping (BPF) decoding, the proposed decoder focuses on prior knowledge of the code bits in the factor graph. Two strategies have been proposed to construct the correction set. The simulation results show that the BPC decoder exhibits a BLER similar to that of BPF decoder while reducing the average latency and complexity by half. In addition, the BLER of the BPC decoder can approach that of the cyclic redundancy check (CRC)-aided SCL (CA-SCL) decoder in regions with low and medium signal-to-noise ratios (SNRs).
For polar codes, bit-flipping strategy can significantly improve the error-correction performance of the belief propagation (BP) decoding. However, the existing BP flipping (BPF) decoding algorithms require many attempts, due to their inefficiency in locating the erroneous positions. This leads to high decoding complexity, limiting their practical applications, especially for long polar codes. In this letter, an enhanced BPF (EBPF) decoding algorithm is proposed. It requires fewer flipping attempts, but yields an advanced decoding performance. By better utilizing the reliability and divergence of bit estimations, a novel weighted metric (WM) is introduced to identify the possible erroneous positions. During each new decoding attempt, the EBPF decoding takes turns to flip these positions. Our simulation results show that, with the same number of flipping attempts, the proposed decoding outperforms the generalized BPF (GBPF) decoding and the state-of-the-art BP correction (BPC) decoding.
We describe a novel approach to interpret a polar code as a low-density parity-check (LDPC)-like code with an underlying sparse decoding graph. This sparse graph is based on the encoding factor graph of polar codes and is suitable for conventional belief propagation (BP) decoding. We discuss several pruning techniques based on the check node decoder (CND) and variable node decoder (VND) update equations, significantly reducing the size (i.e., decoding complexity) of the parity-check matrix. As a result, iterative polar decoding can then be conducted on a sparse graph, akin to the traditional well-established LDPC decoding, e.g., using a fully parallel sum-product algorithm (SPA). This facilitates the systematic analysis and design of polar codes using the well-established tools known from analyzing LDPC codes. We show that the proposed iterative polar decoder has a negligible performance loss for short-to-intermediate codelengths compared to Arikan's original BP decoder. Finally, the proposed decoder is shown to benefit from both reduced complexity and reduced memory requirements and, thus, is more suitable for hardware implementations.
Recently, both polar codes and low-density parity-check (LDPC) codes have been adopted by 3GPP eMBB scenario. Since both codes exist in one system, it is natural to consider the concatenation scheme of them. In this paper, a merged belief propagation (BP) decoding algorithm for the concatenated codes of polar and LDPC codes is proposed. By jointing factor graphs, this merged algorithm is designed by serially executing the LDPC and polar decoding. According to numerical results, the bit error rate (BER) of this proposed algorithm can achieve a notable improvement for different code rates compared with the conventional separated approach. For (1024, 512) polar code, the performance gain of the proposed scheme is at least 0 5 dB even at a low BER of 1 × 10-2. Finally, further refinements of computational complexity and hardware implementation are also provided in this paper.
The Belief Propagation (BP) algorithm has the advantages of high-speed decoding and low latency. To improve the block error rate (BLER) performance of the BP-based algorithm, the BP flipping algorithm was proposed. However, the BP flipping algorithm attempts numerous useless flippings for improving the BLER performance. To reduce the number of decoding attempts needed without any loss of BLER performance, in this paper a metric is presented to evaluate the likelihood that the bits would correct the BP flipping decoding. Based on this, a BP-Step-Flipping (BPSF) algorithm is proposed which only traces the unreliable bits in the flip set (FS) to flip and skips over the reliable ones. In addition, a threshold β is applied when the magnitude of the log–likelihood ratio (LLR) is small, and an enhanced BPSF (EBPSF) algorithm is presented to lower the BLER. With the same FS, the proposed algorithm can reduce the average number of iterations efficiently. Numerical results show the average number of iterations for EBPSF-1 decreases by 77.5% when N = 256, compared with the BP bit-flip-1 (BPF-1) algorithm at Eb/N0 = 1.5 dB.
Since the inherent serial nature of successive cancellation list (SCL) decoding results in a long latency, belief propagation (BP) decoding for polar codes has drawn attention for high-throughput applications. However, its error correction performance is inferior to that of SCL decoding. Therefore, the bit-flipping strategy has been recently applied to BP decoding, which can approach the SCL decoding performance through multiple additional decoding attempts. The original BP flip (BPF) decoding suffers from an inaccurate identification of erroneous bits by a fixed flip set (FS), which has been improved by the generalized BPF (GBPF) decoding. In this article, the GBPF decoding is extended to support multiple bits being flipped in one decoding attempt. In addition, for two types of decoding errors: detected errors and undetected errors, we propose two novel methods to more effectively identify erroneous bits. For detected errors, the concept of loop sets is defined and a loopbased identification method is introduced based on the study of error patterns of BP decoding. On the other hand, a method to generate a more accurate fixed FS is proposed for undetected errors, which considers the bit error distribution under BP decoding. Combining the two methods, the GBPF with merged sets (GBPF-MS) decoding can achieve the SCL-8 performance and outperforms the state-of-the-art BPF, BP list, and SC flip (SCF) decoding, for polar codes with length 1024 and information rate 1/2. Implemented by 40nm CMOS technology, the proposed GBPF-MS decoder with ten flips exhibits an average throughput of 4.19 Gbps at 2.5 dB, which is 1.6× and 1.72× faster than the state-of-the-art SCL-4 and SCF decoders, respectively.