# Improving Data Communication of Enhanced Loran Systems Using 128-_ary_ Polar Codes
## 3 Results
### Comparative Error Performance
This substantial gain underscores the superior error-correcting capability of polar codes when coupled with an effective soft-decision decoding algorithm like SCL, which inherently benefits from the soft inputs provided by the SS demodulator. As shown in Figure 4, while the binary polar code employing the BICM scheme with SS demodulation surpasses the performance of the RS code with EPD-MD, it is still significantly outperformed by the proposed 128-_ary_ polar code with SS demodulation. The BICM scheme is fundamentally suboptimal over the AWGN channel due to its separation of coding and modulation processes [32]. By computing LLRs for each bit independently, the demodulation fails to capture the joint probabilistic information present within a single modulation symbol. In contrast, the proposed 128-ary polar code adopts a non-binary framework that is intrinsically integrated with the modulation scheme. The decoder processes symbol-level likelihoods directly, thereby retaining the complete channel information. This unified design eliminates the information loss characteristic of BICM. This is the main source of the observed performance gain. This suggests that directly designing a 128-_ary_ polar code for the 128-_ary_ PPM, as is carried out in this work, is more effective than using a binary polar code with BICM. Subsequent simulations further demonstrate that the proposed 128-_ary_ polar coding scheme significantly enhances data communication efficiency. Figure 5 illustrates scenarios \begin{table}
\begin{tabular}{c c} \hline \hline
**New Scheme** &
\begin{tabular}{c} **Error Performance Gain at 0.01 FER Level** \\ **(Compared with RS and EPD-MD)** \\ \end{tabular} \\ \hline Correlation receiver & 1.2 dB \\ Soft-decision demodulation & 2.1 dB \\
128-_ary_ polar code with SDD & 6 dB \\ Total & 9.3 dB \\ \hline \hline \end{tabular}
\end{table} 
Table 2: The error performance gain of the proposed 128-_ary_ polar code with SS demodulation. Figure 4: Error performance comparisons between the 128-_ary_ polar code with SS demodulation, the binary polar code with BICM, and the RS code with various demodulations. (with parameters detailed in Table 3) where the proposed system maintains an FER performance that is comparable to the baseline RS code with EPD-MD, but under significantly more favourable transmission parameters, as follows:  
* Specifically, when maintaining the original 56-bit message length, using the 128-_ary_ polar code with SS demodulation allows the number of GRIs required for transmission to be reduced by 2/3 (from 30 to 10 GRIs), while maintaining a similar FER performance. * Alternatively, for a fixed transmission duration of 30 GRIs, the number of message bits can be increased by 250% (from 56 to 175 bits) while achieving a similar FER to the conventional system. * These results underscore the potential of the proposed scheme to substantially increase data communication efficiency, enhancing the communication capabilities of the eLoran system. To elucidate the critical role of soft-decision decoding (SDD) in achieving the observed gains with polar codes, an additional simulation was conducted. Figure 6 compares the 128-_ary_ polar code (30 GRI and 56 bits) using SH demodulation with the RS code also using SH demodulation. In this setup, the SH demodulator provides hard decisions on symbols to the SCL decoder, effectively creating an HDD scenario for the 128-_ary_ polar code. The results indicate that when constrained to HDD, the 128-_ary_ polar code performs approximately 1.6 dB worse than the RS code. This finding strongly suggests that the substantial performance benefits of the 128-_ary_ polar code are intrinsically linked to its effective utilization of soft information through SDD (specifically, SCL decoding). Figure 5: Error performance comparisons between the 128-_ary_ polar code with SS demodulation (varying the GRI number and message bit number) and the RS code with EPD-MD demodulation.