# Improving Data Communication of Enhanced Loran Systems Using 128-_ary_ Polar Codes
## 3 Results  
### Comparative Error Performance  
This section presents a comprehensive evaluation of the proposed 128-_ary_ polar code, comparing its error performance against the RS code using computer simulation. The  
Figure 3: The CERs of polarized subchannels after Monte Carlo simulations with \(\text{SNR}=2.5\). Frame Error Rate (FER) is used to evaluate the error performance under various SNRs over the AWGN channel. The simulations are conducted under ideal synchronization conditions and the number of frames used for the simulation is \(10^{6}\). The parameters in the simulation are presented in Table 1, and the SNR is defined as follows:  
\[\text{SNR}=10\cdot log_{10}\frac{P_{s}}{P_{n}}. \tag{19}\]  
where \(P_{s}\) is the signal power within a 20 kHz bandwidth, and \(P_{n}\) is the noise power within the same bandwidth. According to the authors of [31], the signal power \(P_{s}\) is calculated as \(P_{s}=\left(0.506\ *\ A_{mp}\right)^{2}\), where \(A_{mp}\) is the peak amplitude of the eLoran pulse. For a broader comparison, a binary polar code with the BICM [21] scheme for 128-_ary_ PPM is also simulated. In the BICM scheme, the codeword of the binary polar code is interleaved and divided into 7-bits groups, and each 7-bits group is mapped to a 128-_ary_ PPM symbol. The simulation results, depicted in Figure 4, compare the FER performance of the proposed 128-_ary_ polar code with SS demodulation against the conventional RS code paired with various demodulation techniques. Taking both complexity and error performance into account, we select the 128-_ary_ polar code with \(L=4\) as the proposed scheme. This choice is based on the observation from Figure 4 that increasing \(L\) from 2 to 4 yields a notable gain of approximately 0.4 dB. Since the calculation complexity is linearly related to the list size \(L\) (as discussed in Section 4), further increasing \(L\) to 8 doubles the calculation complexity while providing a negligible gain of approximately 0.2 dB. Thus, \(L=4\) represents an efficient trade-off. Notably, the proposed scheme demonstrates a substantial performance advantage, achieving an SNR gain of approximately 9.3 dB at the 0.01 FER level over the conventional RS code with EPD-MD demodulation--a widely adopted baseline for eLoran receivers--at representative FERs. A detailed breakdown of this cumulative 9.3 dB gain at the 0.01 FER level, also summarized in Table 2, reveals contributions from the following different system enhancements:  
* Replacing the EPD-MD demodulator with a correlation receiver (HH demodulation) for the RS code yields an initial gain of approximately 1.2 dB. This improvement is attributed to the optimal nature of the correlation receiver compared to the simpler EPD-MD. * Further incorporating soft demodulation at the first layer (SH demodulation) with the RS code provides an additional gain of approximately 2.1 dB over the HH demodulation. This result highlights the benefit of soft information. \begin{table}
\begin{tabular}{c c c c} \hline \hline  & **128-_ary_ Polar Code & **RS Code** & **Binary Polar Code (BICM)** \\ \hline Encoding Method & Polar Code & RS Code & Polar Code \\ GRI Number & 30 & 30 & 30 \\ Message Bit Number & 56 & 56 & 56 \\ CRC Bit Number & 14 & 14 & 14 \\ Coded Symbol & 128-_ary_ & 128-_ary_ & Binary \\ Modulation & 128-_ary_ PPM & 128-_ary_ PPM & BICM+128-_ary_ PPM \\ Demodulation & SS/SH & SH/HH/EDP-MD [8] & SS \\ Decoding & SCL & BM & SCL \\ List size of Decoding & 1/2/4/8 & N/A & 4 \\ Number of Frames & \(10^{6}\) & \(10^{6}\) & \(10^{6}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Simulation parameters with the 128-_ary_ polar code, the RS code, and the binary polar code. * The most significant contribution, approximately 6.0 dB, arises from replacing the RS code (with SH demodulation) with the proposed 128-_ary_ polar code utilizing SCL decoding.