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International Conference on Wireless Communications and Signal Processing (WCSP)

Improved Decoder by Partially Permuted Factor Graphs of Polar Codes

Hongji Cui, Kai Niu, Junwei Ren

Published Nov 1, 2022• 1 citation
Publisher

Abstract

We study the structure of the factor graph of polar codes, and propose some improved decoding algorithms for polar codes based on the equivalence of partially permuted factor graphs (PPFG). Different from the previously proposed permuted factor graphs, PPFG proposed in this paper divides the polar code into shorter codewords, each sub-code is permuted into a different factor graph, and finally combined into an equivalent PPFG. We first prove the equivalence of PPFG. Then an algorithm for searching optimal factor graphs is proposed. Finally, we propose a series of algorithms based on PPFG such as optimal PPFG SC/BP, SC parallel list (SCPL) and parallelly permuted SC list (PPSCL). Simulation results show that under the same complexity and delay, the performance of the optimal PPFG SC has a O.25dB gain compared with the original SC decoding algorithm at the code length$N=64$. Algorithms such as SCPL have similar performance to SCL, but SCPL does not require sorting and path expansion.

DOI
10.1109/wcsp55476.2022.10039420

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We study the structure of the factor graph of polar codes, and propose some improved decoding algorithms for polar codes based on the equivalence of partially permuted factor graphs (PPFG). Different from the previously proposed permuted factor graphs, PPFG proposed in this paper divides the polar code into shorter codewords, each sub-code is permuted into a different factor graph, and finally combined into an equivalent PPFG. We first prove the equivalence of PPFG. Then an algorithm for searching optimal factor graphs is proposed. Finally, we propose a series of algorithms based on PPFG such as optimal PPFG SC/BP, SC parallel list (SCPL) and parallelly permuted SC list (PPSCL). Simulation results show that under the same complexity and delay, the performance of the optimal PPFG SC has a O.25dB gain compared with the original SC decoding algorithm at the code lengthN=64N=64N=64. Algorithms such as SCPL have similar performance to SCL, but SCPL does not require sorting and path expansion.
Reed-Muller (RM) and polar codes are a class of capacity-achieving channel coding schemes with the same factor graph representation. Low-complexity decoding algorithms fall short in providing a good error-correction performance for RM and polar codes. Using the symmetric group of RM and polar codes, the specific decoding algorithm can be carried out on multiple permutations of the factor graph to boost the error-correction performance. However, this approach results in high decoding complexity. In this paper, we first derive the total number of factor graph permutations on which the decoding can be performed. We further propose a successive permutation (SP) scheme which finds the permutations on the fly, thus the decoding always progresses on a single factor graph permutation. We show that SP can be used to improve the error-correction performance of RM and polar codes under successive-cancellation (SC) and SC list (SCL) decoding, while keeping the memory requirements of the decoders unaltered. Our results for RM and polar codes of length 128 and rate 0.5 show that when SP is used and at a target frame error rate of 10-4, up to 0.5 dB and 0.1 dB improvement can be achieved for RM and polar codes respectively.
Polar codes are a channel coding scheme for the next generation of wireless communications standard (5G). The belief propagation (BP) decoder allows for parallel decoding of polar codes, making it suitable for high throughput applications. However, the error-correction performance of polar codes under BP decoding is far from the requirements of 5G. It has been shown that the error-correction performance of BP can be improved if the decoding is performed on multiple permuted factor graphs of polar codes. However, a different BP decoding scheduling is required for each factor graph permutation which results in the design of a different decoder for each permutation. Moreover, the selection of the different factor graph permutations is at random, which prevents the decoder to achieve a desirable error correction performance with a small number of permutations. In this paper, we first show that the permutations on the factor graph can be mapped into suitable permutations on the codeword positions. As a result, we can make use of a single decoder for all the permutations. In addition, we introduce a method to construct a set of predetermined permutations which can provide the correct codeword if the decoding fails on the original permutation. We show that for the 5G polar code of length 1024, the error-correction performance of the proposed decoder is more than 0.25 dB better than that of the BP decoder with the same number of random permutations at the frame error rate of 10-4.
Polar codes are provably capacity-achieving errorcorrecting codes suitable for error correction and source coding, constrained coding, and multiple access channels. Many decoding schemes have been proposed for Polar codes, including CRCaided successive cancellation list decoding (CA-SCL), known for its superior error-correcting performance. Belief propagation (BP) techniques have also been explored for decoding polar codes. BP decoding with partially permuted factor graphs (PPFG) exhibits good performance despite being inferior to CA-SCL decoding. Guess Random Additive Noise Decoding (GRAND) is a general decoding scheme for linear codes. Soft-GRAND (SGRAND) exhibits excellent performance for Polar codes despite its high time complexity. We introduce a novel decoding scheme that combines BP decoding with PPFG and SGRAND, advancing the state-of-the-art in Polar code decoding. Our proposed scheme, which fuses BP decoding with PPFG and SGRAND, demonstrates performance similar to that of CA-SCL decoding, representing a significant step forward in Polar code decoding.
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.
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.
Polar codes have become the channel coding scheme for control channel of enhanced mobile broadband in the 5G communication systems. Belief propagation (BP) decoding of polar codes has advantages of low decoding latency and high parallelism but achieves worse bit error ratio (BER) performance compared with the successive cancellation list (SCL) decoding scheme. In this paper, an improved BP list (IBPL) decoding algorithm is proposed with comparable BER performance to SCL algoritm. Firstly, the optimal permuted factor graph is analyzed for polar codes, which improves the performance of the BP decoder without path extension. Furthermore, based on the optimal graph, the bit metric and decoding path metric are proposed to extend and prune the decoding path. The proposed IBPL decoder is focused on not only the permutation of polar codes but also the reliabilities of decoded codewords during each iteration of BP decoding, which has a more accurate decoding path list. The simulation results show that the proposed IBPL decoder improves the BER performance compared with the original BP decoder significantly, and can approach the performance of the SCL decoder at low signal to noise ratio regions.