# Design of A Parallel Decoding Method for LDPC Code Generated via Primitive Polynomial
## 4 Numerical Results  
In this section, we simulate the BER of m sequence codes generated via primitive polynomials \(f(x)=x^{89}+x^{38}+1\) and \(f(x)=x^{33}+x^{13}+1\). The code rates of these two codes are the same \(0.03\) and the code lengths are \(3000\) and \(1100\), respectively. There are two decoding methods employed in the simulation, one is the proposed parallel decoding method with multiple sub-decoders, the other one is the single-decoder decoding method which employed in [1]. The channel is additive white Gaussian noise (AWGN) channel.  
Firstly, we simulate the BER performance of the proposed decoding method with different number of sub-decoders. According to the construction method of parity-check matrix proposed in Section 3.1, we have found the sampling intervals \(q=3,5,7,9,11,13\) to sample the m-sequence generated by \(f(x)=x^{89}+x^{38}+1\) and obtain the primitive polynomials of the sampling sequences.  
\[\begin{cases}f_{3}(x)=x^{89}+x^{72}+x^{55}+x^{38}+1\\ f_{5}(x)=x^{89}+x^{61}+x^{38}+x^{33}+1\\ f_{7}(x)=x^{89}+x^{69}+x^{38}+x^{29}+1\\ f_{9}(x)=x^{89}+x^{72}+x^{55}+x^{38}+x^{31}+x^{24}+1\\ f_{11}(x)=x^{89}+x^{67}+x^{52}+x^{38}+x^{30}+x^{15}+1\\ f_{13}(x)=x^{89}+x^{44}+x^{43}+x^{40}+x^{39}+x^{38}+1\end{cases}. \tag{14}\]  
For the m-sequence generated by \(f(x)=x^{33}+x^{13}+1\), we also use sampling intervals \(q=3,5,7,9\) to get the sampling sequences and corresponding primitive polynomials.  
\[\begin{cases}f_{3}(x)=x^{33}+x^{29}+x^{17}+x^{13}+1\\ f_{5}(x)=x^{33}+x^{22}+x^{13}+x^{11}+1\\ f_{7}(x)=x^{33}+x^{16}+x^{14}+x^{13}+1\\ f_{9}(x)=x^{33}+x^{22}+x^{16}+x^{13}+x^{11}+x^{8}+1\end{cases}. \tag{15}\]  
Then we have simulated the BER performance of the proposed decoding method with different number of sub-decoders for these two codes, the results are shown in Figure 4.  
Figure 4 presents the relation between BER performance and the number of sub-decoders. With the increase of employed sub-decoders, the BER performance is better but the computation complexity is increased. In our method, all the sub-decoders have nearly the same computation complexity. Therefore, the parallel decoding method with \(N\) sub-decoders is \(N\) times complex than the original single-decoder system. To get the tradeoff of BER performance and complexity, we employ 5 sub-decoders for the code with length 3000 as Figure 4a has shown that the BER curve is converged when employing 5 sub-decoders. Similarly, we employ 4 sub-decoders for the code with length 1100.  
In the end, we have simulated the BER performance of these two codes with different decoding methods, the results are shown in Figure 5.  
From Figure 5 we can see that, the proposed decoding method outperforms the single-decoder decoding method in terms of BER when the value of \(E_{b}/N_{0}\) is fixed. The \(E_{b}/N_{0}\) range of the simulation is from 1.4 dB to 4.2 dB while the corresponding SNR range is from \(-13.82\) dB to \(-11.02\) dB. In the BER region of \(10^{-5}\), our decoding method has a BER gain of 0.3 dB and 0.4 dB compared to the single-decoder decoding method for these two codes, respectively.  
Figure 4: The bit error rate (BER) performance of the proposed decoding method with different number of sub-decoders. (**a**) code length 3000. (**b**) code length 1100.