Based on pruning techniques that interpret polar codes as low-density parity-check-like (LDPC-like) codes, this paper develops novel belief propagation (BP)-based decoders utilizing the concept of schedule diversity. By analyzing the impact of hidden variable nodes (VNHs), scheduling algorithms that guarantee fast convergence speeds and superior error performance are devised for the LDPC-like sparse factor graph (SPF), incorporating both standard sequential scheduling (SSS) and informed dynamic scheduling (IDS) algorithms. To further enhance error performance, cyclic redundancy check (CRC) bits are integrated into the LDPC-like SPF, and random update sequences are introduced to reduce computational complexity while implementing schedule diversity. Inspired by belief propagation list decoding, the proposed approaches eventually update messages in the LDPC-like SPF according to various schedules, thereby producing diverse decoding outcomes. Compared to the CRC-aided successive cancellation list (CA-SCL) decoder and benchmark soft-output decoders in the literature, the proposed schemes achieve comparable or improved error performance while reducing computational complexity in the high SNR region. These results serve as preliminary findings on the BP algorithm over LDPC-like SFGs and provide useful insights for the design and practical improvement of polar decoders.
Based on pruning techniques that interpret polar codes as low-density parity-check-like (LDPC-like) codes, this paper develops novel belief propagation (BP)-based decoders utilizing the concept of schedule diversity. By analyzing the impact of hidden variable nodes (VNHs), scheduling algorithms that guarantee fast convergence speeds and superior error performance are devised for the LDPC-like sparse factor graph (SPF), incorporating both standard sequential scheduling (SSS) and informed dynamic scheduling (IDS) algorithms. To further enhance error performance, cyclic redundancy check (CRC) bits are integrated into the LDPC-like SPF, and random update sequences are introduced to reduce computational complexity while implementing schedule diversity. Inspired by belief propagation list decoding, the proposed approaches eventually update messages in the LDPC-like SPF according to various schedules, thereby producing diverse decoding outcomes. Compared to the CRC-aided successive cancellation list (CA-SCL) decoder and benchmark soft-output decoders in the literature, the proposed schemes achieve comparable or improved error performance while reducing computational complexity in the high SNR region. These results serve as preliminary findings on the BP algorithm over LDPC-like SFGs and provide useful insights for the design and practical improvement of polar decoders.
In this paper, based on the technique of LDPC (low-density parity-check)-like sparse graphs for the Polar codes, we propose a Polar decoding method which provides the schedule diversity. The investigated decoding schedules outperforms Arikan's belief propagation (BP) algorithm performed on the original encoding factor graph. Secondly, based on both the layered BP (LBP) and the modified node-wise residual BP (NW-RBP) schedules, we are able produce several sequential decoding schedules that are executed in serial. It is shown through simulations that when compared to the successive cancellation list (SCL) decoding, the proposed scheme achieves a lower complexity level, while a gap of 0.2 dB still exists in BER performance in the high-SNR region. However, the enhanced BP-based decoding method provides a flexible decoding latency, and is able to generate soft-in/soft-out information, which makes it suitable for the schemes which perform iterative decoding and detection processes.
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).
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.
To improve the performance of BP decoding for polar codes, researchers have developed a Belief Propagation List (BPL) decoding algorithm by executing multiple BP decoders in parallel.This algorithm outperforms traditional BP decoding based on single graph decoding, although it sacrifices some complexity and delay.Unlike this basic Polar-BPL decoding mentioned above, a low-density parity check like (LDPC-like) BPL decoding algorithm has been introduced into polar codes.This algorithm performs more efficient BP decoding iterations on sparse graphs.To further reduce the complexity and delay of LDPC-like BPL, and avoid analyzing all sparse graphs, this article proposes an improved LDPC-like BPL (ILDPC-like BPL) decoding algorithm based on selecting factor graphs.This algorithm analyzes differences in permuted factor graphs and selects these graphs with non-convergent decoding results to form a decoding list, allowing decoding to be completed with fewer graphs traversed and iterations, thereby reducing complexity and delay.On this basis, to improve the reliability of this algorithm, this article further adds a path metric (PM) mechanism, which can dynamically evaluate different sparse graphs' results in parallel decoding.The simulation results show that compared with LDPC-like BPL (random scheme), ILDPC-like BPL improves decoding performance while reducing decoding delay and complexity; Compared with basic Polar-BPL (cycle-shift scheme), this algorithm greatly reduces decoding delay and complexity while achieving similar performance.
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).
Owing to its high parallelism, belief propagation (BP) decoding is highly amenable to high-throughput implementations and thus represents a promising solution for meeting the ultra-high peak data rate of future communication systems. However, for polar codes, the error-correcting performance of BP decoding is far inferior to that of the widely used CRC-aided successive cancellation list (SCL) decoding algorithm. To close the performance gap to SCL, BP list (BPL) decoding expands the exploration of candidate codewords through multiple permuted factor graphs (PFGs). From an implementation perspective, designing a unified and flexible hardware architecture for BPL decoding that supports various PFGs and code configurations presents a big challenge. In this paper, we propose the first hardware implementation of a BPL decoder for polar codes and overcome the implementation challenge by applying a hardware-friendly algorithm that generates flexible permutations on-the-fly. First, we derive the graph selection gain and provide a sequential generation (SG) algorithm to obtain a near-optimal PFG set. We further prove that any permutation can be decomposed into a combination of multiple fixed routings, and we design a low-complexity permutation network to satisfy the decoding schedule. Our BPL decoder not only has a low decoding latency by executing the decoding and permutation generation in parallel, but also supports an arbitrary list size without any area overhead. Experimental results show that, for length-1024 polar codes with a code rate of one-half, our BPL decoder with 32 PFGs has a similar error-correcting performance to SCL with a list size of 4 and achieves a throughput of 25.63 Gbps and an area efficiency of 29.46 Gbps/mm2 at SNR=4.0dB, which is 1.82× and 4.33× faster than the state-of-the-art BP flip and SCL decoders,~respectively