## 1 Introduction  
Polar codes [1] have been widely studied since they were proposed due to their excellent performance. Consequently, polar codes have been selected as the control channel coding scheme in the 5G enhanced Mobile Broadband (eMBB) scenario [2] and are expected to still be competitive coding technologies in the future. In particular, polar codes using a successive cancellation (SC) decoder comprise the first channel coding technique that has been proven to achieve channel capacity with an infinite code length [1]. However, the error-correction performance of polar codes using the SC decoder with limited code length is unsatisfactory. Therefore, improving the performance of polar codes with limited code length has always been a concern of scholars, and designing an efficient decoder is one of the most effective ways to solve this problem. The SC list (SCL) [3, 4] and CRC-aided SCL (CA-SCL) [5, 6] are two improved list decoders based on the SC decoder. Compared with the single candidate path of the SC decoder, they adopt a list of multiple candidate paths and posterior probability to improve the error-correction performance. In particular, the CA-SCL decoder can utilize CRC to filter wrong candidate paths on the list and obtain better performance than the SCL decoder. However, this list containing multiple candidate paths requires more computational complexity (energy) and storage space. To further optimize the CA-SCL decoder, many scholars have proposed different improved list decoders. The authors of [7] proposed a new path metric that is beneficial to hardware implementation for the filtering of candidate paths and is widely used nowadays. [PERSON]. [8] combined the adaptive list and CRC to further reduce the average complexity of the CA-SCL decoder without loss of error-correction performance. In [9, 10], the authors designed different distributed CRCs for the early termination of the CA-SCL decoder. The authors of [11] designed a threshold to reduce redundant calculations during list decoding. Furthermore, [PERSON]. [12] proposed a polar decoder supported by parity check (PC), which has an error-correction performance comparable to that of the CA-SCL decoder. Although the list decoders above improve the performance of the CA-SCL decoder to a certain extent, their error-correction performance is similar to or even lower than that of the CA-SCL decoder with the same list size. In addition to list decoders, [PERSON]. [13] proposed the first flip decoder called SC flip (SCFlip). The core idea of this flip decoder is to use additional SC decoding attempts to perform flip operations in the error-prone positions, and thus the flip decoder can achieve better error-correction performance than the SC decoder. Subsequently, many scholars have contributed to improving the performance of the flip decoder. Ref. [14] effectively obtained the error-prone positions to enhance the efficiency of flip operations. Ref. [15] designed the DSCFlip decoder, which employs new flip metric to correct high-order errors and further improve the error-correction performance of flip decoders. Ref. [16] simplified the flip metric of the DSCFlip decoder to improve its application value. Moreover, Ref. [17] first introduced parity-check bits into flip decoders to obtain a more efficient early termination capability. Their advantages are that they can achieve the performance of the CA-SCL decoder by using only a single path, and their average computational complexity is close to that of the SC decoder at a high signal-to-noise ratio (SNR). However, the computational complexity of these flip decoders may be higher than that of the CA-SCL decoder with the same error-correction performance at a low SNR. Reference [18] proposed the first list flip decoder called the SCL bit-flip (SCL-BF) decoder. This decoder creates a bit-flip operation in list decoding processes, which combines the flip idea in flip decoders and the list in list decoders; it then realizes a good performance trade-off between the list decoders and the flip decoders.