# Improving Data Communication of Enhanced Loran Systems Using 128-_ary_ Polar Codes
## 4 Discussion  
The simulation results presented herein compellingly demonstrate the substantial benefits of adopting the proposed 128-_ary_ polar codes with SS demodulation for the eLoran data communication system. This approach offers significant advantages in error performance and data transmission efficiency compared to the conventional RS code. Specifically, our findings highlight two key advantages:  
1. Enhanced Error Performance: For equivalent message lengths and the number of GRIs, the 128-_ary_ polar code with SS demodulation achieves an approximate 9.3 dB SNR gain over the conventional RS code with EPD-MD demodulation. 2. Improved Data Communication Efficiency: Alternatively, while maintaining a comparable error performance to the traditional system, the proposed scheme can reduce the required number of GRIs by 2/3 (from 30 to 10 for a 56-bit message) or increase the message bit number by 250% (from 56 to 175 bits within 30 GRIs). A systematic analysis of the simulations and comparisons reveals the contributing factors to this performance gain. The adoption of a correlation receiver (HH demodulation) provides an initial improvement over EPD-MD demodulation. The introduction of soft-decision demodulation (transitioning from HH to SH demodulation for the RS code) further enhances performance by preserving valuable channel information. However, the most significant gain is realized through the combination of the 128-_ary_ polar code with SDD, specifically using the SCL algorithm. Our results (Figure 6) underscore this point: when the 128-_ary_ polar code is constrained to hard-decision inputs (via SH demodulation), its performance degrades, even falling below that of the RS code under similar hard-decision conditions. While one might consider applying SDD techniques to the RS code, the practical challenges are considerable. As noted in [15], the optimal Maximum Likelihood (ML) decoding of RS codes is an NP-hard problem, rendering it computationally infeasible for real-time systems. Existing sub-optimal SDD algorithms for RS codes typically offer only modest gains over HDD and often come with a high complexity penalty. In contrast, the 128-ary polar code with SCL decoding offers a practical and effective SDD solution with acceptable  
Figure 7: Error performance comparison between the eLoran system in the absence of Doppler shift and under a 2.5 Hz Doppler shift. complexity, particularly suited to the short-code nature of eLoran data transmissions. In this paper, we employ the decoding method introduced in [23], which primarily results in a storage complexity of \(O\left(2\cdot q^{2}\right)\) and a calculation complexity of \(O\left(L\cdot N\cdot log_{2}N\cdot q^{2}\right)\). Given the substantial performance gains previously discussed, the complexity is considered both acceptable and feasible for implementation in short-code eLoran systems. The 128-_ary_ polar code presented in this paper successfully integrates these advantages into the eLoran system. A crucial aspect of the proposed scheme is its compatibility with the existing eLoran infrastructure. By retaining 128-_ary_ PPM, which is fundamental to the eLoran system's positioning function, our approach ensures that enhancements to data communication capability do not compromise the primary navigation service. In summary, the replacement of conventional RS codes with the proposed 128-_ary_ polar code, supported by SS demodulation and SCL decoding, offers a significant improvement to the eLoran data communication system. The demonstrated gains in error performance and data efficiency are achieved while maintaining compatibility with conventional positioning functions and acceptable computational complexity. Future work may extend the modulation set beyond balanced modulation, optimize symbol mapping through constellation shaping to enhance resilience, and further reduce the complexity of SCL decoding to improve efficiency. Conducting these studies is expected to further improve the error performance of polar codes in eLoran systems. Conceptualization, D.Q.