# Sequential Polar Decoding with Cost Metric Threshold
## 4 Simulation Results  
Let us conduct a comprehensive performance evaluation of the proposed SC-Creeper decoder, comparing it with four other decoding methods: SC, SCL (\(L=8\)), SCS (\(L=8\), \(D=1024\)) and SC-Fano. The evaluation focuses on two critical aspects: performance and normalized complexity. To accurately simulate the channels, the authors use AdditiveWhite Gaussian Noise (AWGN [18; 19]) with Binary Phase Shift Keying (BPSK). Performance is evaluated with the Frame Error Rate (FER) metric. For CRC checking, we utilize a polynomial with an additional 16 bits, defined as \(g(x)=x^{16}+x^{15}+x^{2}+1\). A CRC code with 16 bits provides a higher level of error detection compared to shorter CRC codes. This is important in polar decoding because it helps to identify and correct errors more effectively. Moreover, it is a common choice in many communication standards and protocols. In this work, we study the 5G NR strategy [20] to perform polar codes and to construct a reliable set. In the case of SC-Creeper, the parameter \(\Delta=1\) is chosen, so movement along the tree (in both directions) occurs in steps of approximately one vertex. For the additional cost function, the iterative procedure presented below is chosen:  
\[\Gamma(i)= \begin{cases}-\infty,\ i\leq 0,\\ \Gamma(i-1)+2,\ \text{if node is new},\\ i,\ \text{otherwise}.\end{cases}  \tag{4}\]  
Moreover, for the first realization, we assume that \(T\equiv\Gamma\), so the threshold function has the same properties and limitations as a cost function. Next, in Figures 2-4, we provide the simulation result for codes with length \(N=\{256,512,1024\}\), a high rate \(R=\frac{3}{4}\), a moderate rate \(R=\frac{1}{2}\) and a low rate \(R=\frac{1}{4}\). The number of blocks to test the performance of the decoders is chosen to be \(10^{6}\), and the number of maximum failed decoding attempts at one SNR point is equal to 30. Figure 2: Comparison of FER values, \(N=256\). SC-Creeper's performance is comparable to that of SCL with list size \(L=8\). However, using the additional cost metric improves the performance, so the modified SC-Creeper has a stronger correction ability. Thus, the option of using an algorithm without a list with two stacks appears to be more optimal in terms of performance (and also more optimal than the basic SCS decoder, which has only one stack). One can also notice that at low and moderate \(R\leq\frac{1}{2}\), the modified Creeper algorithm, as well as its main version, perform worse than, for example, the SCL-8 decoder. This is due to the fact that a frozen set is growing, for which there is no clear definition of the threshold values and cost functions. It is possible that, with additional regularization at low rates, it will be possible to obtain higher performance, taking into account the modification of the frozen set \(\mathcal{F}\). Comparing the performance of various polar decoding algorithms in relation to the Frame Error Rate (FER) at different \(E_{b}/N_{0}\) values provides valuable insights into their effectiveness across different scenarios. Let us analyze the characteristics and performance of each algorithm. The basic SC algorithm serves as the foundation, offering decent performance but struggling with high code rates and long block lengths due to its sequential nature. SCL algorithms, such as SCL-8, maintain lists of candidate solutions, which is why they are effective across a range of code rates and block lengths, providing a good trade-off between complexity and performance. Figure 3: Comparison of FER values, \(N=512\). SCS aims to simplify the decoding process while maintaining reasonable performance. It may sacrifice some decoding accuracy for reduced complexity, making it suitable for low-complexity applications. SC-Fano introduces Fano metric-based decisions, improving upon SC but still falling short of SCL in some cases. SC-Creeper employs iterative refinement, particularly beneficial for high code rates, with potential complexity reductions compared to SCL.