## 1 Introduction
For brevity, we use the term \"list flip operation\" to refer to the flip operation of a list flip decoder. On the one hand, the SCL-BF decoder can obtain the error-correction performance of the CA-SCL decoder with a large list by utilizing a smaller list. On the other hand, the SCL-BF decoder can use a list of multiple candidate paths to increase the probability of keeping the correct path. However, the list flip operation and the selection of error-prone positions in the SCL-BF decoder are very similar to those in the SCFFlip decoder, which means that it is not efficient enough for correcting errors in a more complex list decoding scheme. To further improve the performance gain brought by list flip operations, [19; 20] proposed different flip metrics to optimize the original selection of error-prone positions, and [21] proposed a list decoder with shift-pruning (SCL-SP) to optimize the original list flip operations in the SCL-BF decoder. Moreover, [PERSON]. [22; 23] proposed different list flip decoders to optimize the shifted-pruning operation in the SCL-SP decoder. However, these list flip decoders cannot effectively flip high-order errors, which limits the error-correction performance potential of list flip decoders. A high-order error means that correcting this error requires multiple flip operations in one decoding attempt. Recently, [PERSON]. [24] proposed a dynamic SCL-Flip (D-SCLF) decoder with a high-order error correction capability. This decoder employs new flip metric to estimate the likelihood of high-order errors in list decoding processes; it can therefore correct high-order errors in these list decoding processes. However, compared to the flip metric in [20], this new flip metric includes additional exponential and logarithmic operations. Furthermore, the number of these exponential and logarithmic operations grows exponentially with the order of error correction (the maximum number of list flip operations in one decoding attempt) and the number of information bits, which then reduces the application value of the list flip decoder with high-order error correction capacity. To further develop the list flip decoder for polar codes, we designed an adaptive list flip (ALF) decoder with high-order error correction capability and a simplified flip metric. The contributions of this decoder are as follows:  
* We designed a new simplified flip metric. This simplified flip metric replaces the additional exponential and logarithmic operations introduced by the original flip metric with multiplication and addition operations. The simulation results show that the D-SCLF decoder using the simplified metric has similar error-correction and complexity performance to the D-SCLF decoder using the original flip metric. * We theoretically analyzed the characteristics of the combination of the adaptive list and list flip decoders, and we verified these theoretical analyses by simulation. * We proposed an ALF decoder with a high-order error correction capability. The decoder mainly combines the simplified flip metrics and the adaptive list. Simulation results show that our ALF decoder can effectively reduce the average computational complexity (energy) of the D-SCLF decoder without diminishing the error-correction performance. The remainder of the paper is organized as follows. Section 2 briefly gives an overview of the polar encoding method and some polar decoding algorithms. Section 3 describes and analyzes the details of the proposed decoders. In Section 4, the simulation results are illustrated and discussed. Finally, some conclusions are highlighted.