# Polar Code Parameter Recognition Algorithm Based on Dual Space
## 4 Numerical Simulation
### Algorithm Complexity  
Setting the amount of intercepted data as Len, when the traversal calculation reaches the code length \(n^{\prime}(n_{\text{min}}\leq n^{\prime}\leq n_{\text{max}})\) and the code rate \(R^{\prime}(R_{\text{min}}\leq R^{\prime}\leq R_{\text{max}})\), the dual matrix \(\mathbf{H}\) is calculated after removing the \(n^{\prime}(1-R^{\prime})\) rows in the construction matrix \(\mathbf{G}_{n^{\prime}}\); then, the statistics AL and thresholds \(\Lambda_{\text{opt}}\) of \(\lfloor\text{Len}/n^{\prime}\rfloor\) codewords are calculated. The dual matrix with fixed code length and rate is fixed, so it can be generated in advance and read directly when needed. The calculation of statistics requires \(\lfloor\text{Len}/n^{\prime}\rfloor\cdot n^{\prime}(1-R^{\prime})\) vector additions, \(\lfloor\text{Len}/n^{\prime}\rfloor\cdot n^{\prime}(1-R^{\prime})\) comparisons, and \(\lfloor\text{Len}/n^{\prime}\rfloor\cdot n^{\prime}(1-R^{\prime})\) vector multiplications. For simplicity, one comparison is equivalent to one vector addition, so the number of addition computations of the parameter recognition algorithm proposed in this paper is \(\sum\limits_{R^{\prime}=R_{\text{min}}}^{R^{\prime}=R_{\text{max}}}\frac{n^ {\prime}=n_{\text{max}}}{n^{\prime}=n_{\text{min}}}^{2}\lfloor\text{Len}/n^{ \prime}\rfloor\cdot n^{\prime}(1-R^{\prime})\) and the number of multiplications is \(\sum\limits_{R^{\prime}=R_{\text{min}}}^{R^{\prime}=R_{\text{max}}}\frac{n^ {\prime}=n_{\text{max}}}{n^{\prime}=n_{\text{min}}}^{n^{\prime}=n_{\text{max}}} \lfloor\text{Len}/n^{\prime}\rfloor\cdot n^{\prime}(1-R^{\prime})\). Therefore, the complexity of the algorithm proposed in this paper approximately increases linearly with increasing length of the intercepted data and decreasing code rate. Regarding the algorithm proposed in [15], the number of multiplications is \(\sum\limits_{n^{\prime}=n_{\text{min}}}^{n^{\prime}=n_{\text{max}}}20\cdot n ^{\prime}\), the number of additions is \(\sum\limits_{n^{\prime}=n_{\text{min}}}^{n^{\prime}=n_{\text{max}}}\text{Len }\cdot n^{\prime}\), and the calculation amount increases approximately linearly with increasing length of the intercepted data and code length. The algorithm proposed in [16] requires at most \(5\cdot\text{Len}+10\cdot n^{\prime}+(2\cdot n^{\prime}-1)\cdot\text{Len}\) multiplications and \(\left(\text{Len}/(2\cdot n^{\prime})-1\right)\cdot\left(2\cdot n^{\prime} \right)^{2}\) additions. Let the sum of the number of multiplications and additions be the complexity, and we show how code length, code rate, and the received sequence length influence the complexity in what follows. When the influence of code length and code rate on complexity is considered, the length of the received sequence length is set to 10,000. When the influence of sequence length on complexity is considered, a code length of 512 and a code rate of 1/2 are chosen.  
As is shown in Figure 12, the complexities of the algorithms proposed in [15; 16] are not affected by code rate so they just have one curve, but the complexity of the algorithm proposed in this paper is affected by code rate, so the complexity with every code rate has a curve. All the complexities increase as the code length increases from 32 to 1024,  
Figure 13: Influence of sequence length on complexity [15; 16].  
but the complexity of the algorithm proposed in this paper increases only when the code rate increases from 1/5 to 8/9. As shown in Figure 13, all the complexities increase as the sequence length increases from 2000 to 10,000. Regardless, the complexity of the algorithm proposed in this paper is the lowest.