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International Symposium on Problems of Redundancy in Information and Control Systems (REDUNDANCY)

Generalized Belief Propagation Decoding of Polar Codes

Nikolai Iakuba, Peter Trifonov

Published Oct 24, 2023• 1 citation
Publisher

Abstract

A generalized belief propagation algorithm is proposed for decoding of polar codes. The proposed approach relies on offline successive transformations of the factor graph, which merge parity check nodes into check nodes corresponding to some non-trivial codes. It was shown, that the constructed factor graph for (2048, 1024) polar code provides 0.25 performance gain under generalized belief propagation decoding with 5 iterations and lower worst case complexity compared to state-of-the-art belief propagation decoders of polar codes.

DOI
10.1109/redundancy59964.2023.10330195

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•Feb 3, 2022•0 citations
A generalized belief propagation algorithm is proposed for decoding of polar codes. The proposed approach relies on offline successive transformations of the factor graph, which merge parity check nodes into check nodes corresponding to some non-trivial codes. It was shown, that the constructed factor graph for (2048, 1024) polar code provides 0.25 performance gain under generalized belief propagation decoding with 5 iterations and lower worst case complexity compared to state-of-the-art belief propagation decoders of polar codes.
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.
We show that the performance of iterative belief propagation (BP) decoding of polar codes can be enhanced by decoding over different carefully chosen factor graph realizations. With a genie-aided stopping condition, it can achieve the successive cancellation list (SCL) decoding performance which has already been shown to achieve the maximum likelihood (ML) bound provided that the list size is sufficiently large. The proposed decoder is based on different realizations of the polar code factor graph with randomly permuted stages during decoding. Additionally, a different way of visualizing the polar code factor graph is presented, facilitating the analysis of the underlying factor graph and the comparison of different graph permutations. In our proposed decoder, a high rate Cyclic Redundancy Check (CRC) code is concatenated with a polar code and used as an iteration stopping criterion (i.e., genie) to even outperform the SCL decoder of the plain polar code (without the CRC-aid). Although our permuted factor graph-based decoder does not outperform the SCL-CRC decoder, it achieves, to the best of our knowledge, the best performance of all iterative polar decoders presented thus far.
Abstract In this letter, a modified message passing algorithm to prevent short cycles in the factor graph is proposed and combined with a relaxation technique to improve the performance of cyclic redundancy check aided belief propagation decoding for polar codes. Numerical results show that the proposed scheme can achieve a 0.7 dB coding gain, for the code rate of 0.5 at the block error rate of 10 −5 with a comparable average number of iterations over the pure belief propagation decoding scheme when the code block length is 1024.
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.
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.
A polar code decoder based on the belief propagation algorithm is desirable because of the potentially low latency and its suitability for parallel execution on multicore and SIMD processors. However, current state-of-the-art algorithms require many iterations to achieve comparable bit and frame error rate compared to successive cancellation algorithms. Also, the current state-of-the-art belief propagation algorithms have a high computational complexity compared to successive cancellation. In this paper we present an enhanced belief propagation algorithm, in which parts of the computations are altered to reduce the negative effect of the short cycles in the polar code factor graph. Our proposed algorithm has a gain of approx+0.4mathbf{dB} in both frame and bit error rate compared to successive cancellation and a gain of approx+0.16mathbf{dB} and approx+0.13mathbf{dB} at a frame and bit error rate of 10-3 respectively, compared to belief propagation. Also, the maximum number of iterations of our algorithm is reduce to 0.6cdot I_{max}. As a result, the latency is up to approx 11 times lower compared to successive cancellation and up to approx 1.8 times lower compared to the current state-of-the-art belief propagation polar code algorithm. Furthermore, the reduction in the maximum iteration count results in a lower power consumption after implementation.