Polar code [
1] is a mathematically proven capacity-achieving coding scheme and has specific construction methods and linear encoding and decoding complexity, which have been selected as the coding scheme in fifth-generation enhanced mobile broadband (eMBB) control channels [
2]. A successive cancellation list (SCL) decoder with a cyclic redundancy check (CRC) (CA-SCL) [
3,
4] obtains comparable error performance with that of the state-of-the-art LDPC code. However, it suffers from high latency due to its intrinsic serial structure. Conversely, the belief propagation (BP) decoder [
5,
6] performs the decoding process in a parallel manner, therefore garnering more and more attention as a low-latency implementation method.
However, the original BP decoder has a worse block error rate (BLER) performance than that of the SCL decoder. Therefore, several BP-based decoding algorithms have been proposed to enhance BLER performance. By considering different representations obtained using graph permutations, the multi-trellis scheme, first proposed in [
7], was further improved in [
8,
9,
10] with different permuted patterns, resulting in better BLER performance, but its throughput subsequently dropped. In [
11], a BP list (BPL) decoder performed
L independent BP decoders on different permuted versions of the factor graph to yield a valid estimate. Additionally, the selection strategy of
L permuted factor graphs which have the least number of girths was designed in [
12], resulting in better performance and lower average decoding latency. Post-processing is another effective strategy to improve the BLER performance. A random perturbation-based post-processing method was developed in [
13] to address different types of BP errors and to improve convergence. The noise-aided BPL decoder, combined with random noise injection and list decoding, introduced in [
14], has a much better performance than the SCL and is a bit behind the CA-SCL decoder in terms of BLER performance. Inspired by the SC bit-flipping decoder, the BP flipping (BPF) decoder, as a post-processing method, was first proposed in [
15] and was then improved in [
16,
17,
18]. The BPF decoder, performing bit-flipping on error-prone information bits, achieved a comparable BLER performance with that of the SCL decoder with a small list size. The flipping set which is constructed by a convolutional neural network-based imitation learning scheme was presented in [
19], such that useless flipping attempts are avoided. [
20] proposed a noised-aided BPL bit-flipping decoder which is superior to the CA-SCL decoder at the cost of numbers of decoding iterations. In [
21], a low-complexity two-level post-processing algorithm was proposed to modify the false converged error efficiently in the BPL decoder. In [
22], the correction performed on code bits was studied. When the BP decoder failed, the correction operation performed as an external perturbation used to improve the convergence, named the BPC decoder. By correcting the unreliable code bits, i.e., assigning
to their log-likelihood rate (LLR) and performing additional iterations, the decoder is expected to converge correctly. For identical additional decoding attempts, the BPC decoder shows better BLER performance than the BPF decoder. However, without benefiting from prior knowledge about the identified code bit, assignments on both sides lead to excessive decoding attempts, resulting in relatively high decoding latency overhead.
In this article, the BPC decoder is modified by redefining the correction set based on a new metric and the correction operation based on the predefined threshold, abbreviated as the MBPC decoder. Then, the proposed MBPC decoding is extended to a higher-order scheme, named MBPC-, where code bits are corrected in one decoding attempt. The simulation results show that the MBPC decoder shows a slight performance gain but has less complexity and latency than the original BPC decoder. The higher-order MBPC decoders are verified to considerably improve the BLER performance. For polar codes of length with 24-bit CRC termination, when , the MBPC-2 decoder achieves around a dB performance gain compared with the original BP decoder. Compared with existing BP-based multiple flipping decoders, the MBPC-2 performs a similar BLER but approximately half iterations. At , the MBPC-2 decoder achieves a gain of dB compared with the CA-SCL decoder with , at a latency overhead saving of . The contributions of this article are summarized as follows.