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International Symposium on Turbo Codes and Iterative Information Processing

Improved Hybrid RM-Polar Codes and Decoding on Stable Permuted Factor Graphs

Wu Wei, Zhen Zhai, Paul H. Siegel

Published Aug 30, 2021• 3 citations
Publisher

Abstract

A new family of modified polar codes and a new permutation selection scheme for belief propagation list (BPL) decoding are presented. We first propose a new code construction methodology to interpolate between Reed-Muller (RM) codes and polar codes. By taking advantage of an existing partial order on bit-channels whose corresponding indices share the same Hamming weight, we analyze the complexity of the new construction method. It is shown that we need to compute the reliability of roughly a fraction 1/log3/2N of all the bit-channels contained in the subset. Then, we explore a special family of factor-graph layer permutations called stable permutations (SPs) that preserve a specified information set when the corresponding bit permutations are applied to message bit indices. Simulation results show that the error-rate performance of the new family of codes is better than that of 5G polar codes under successive cancellation list (SCL) decoding. In addition, the SP selection scheme stands out as a preferred one for BPL decoding in terms of error-rate performance, while the average number of iterations per belief propagation (BP) decoder on the permuted factor graph is close to that of the original BP decoder when an early stopping condition is applied.

DOI
10.1109/istc49272.2021.9594172

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A new family of modified polar codes and a new permutation selection scheme for belief propagation list (BPL) decoding are presented. We first propose a new code construction methodology to interpolate between Reed-Muller (RM) codes and polar codes. By taking advantage of an existing partial order on bit-channels whose corresponding indices share the same Hamming weight, we analyze the complexity of the new construction method. It is shown that we need to compute the reliability of roughly a fraction 1/log3/2N of all the bit-channels contained in the subset. Then, we explore a special family of factor-graph layer permutations called stable permutations (SPs) that preserve a specified information set when the corresponding bit permutations are applied to message bit indices. Simulation results show that the error-rate performance of the new family of codes is better than that of 5G polar codes under successive cancellation list (SCL) decoding. In addition, the SP selection scheme stands out as a preferred one for BPL decoding in terms of error-rate performance, while the average number of iterations per belief propagation (BP) decoder on the permuted factor graph is close to that of the original BP decoder when an early stopping condition is applied.
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.
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.
The belief propagation (BP) decoding of polar codes provides better bit error rate (BER) and packet error rate (PER) performance than the successive cancelation (SC) decoding. It has been reported in the literature that the graph of BP decoding of polar codes can be permuted to provide even better decoding performance. In this paper, we theoretically prove that all permutations of the graph are equivalent in terms of the polar encoding. For BP decoding, it is shown that permutations of the layers of the graph also permute the bit channels. A principle which protects the weakest information bit channel is provided in the paper. Two parallel BP decoders, one employing the standard graph and the other selected based on the selection principle, are also proposed to work together. The simulation results show that when the total number of iterations is fixed, the proposed BP decoding can have the same performance as the existing permutation scheme while provides a higher throughput.
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^{2}2 at SNR=4.0dB, which is 1.82×\times× and 4.33×\times× faster than the state-of-the-art BP flip and SCL decoders,~respectively
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.