With the rapid development of Global Navigation Satellite System (GNSS) technology, precise positioning and navigation have become increasingly important in a wide range of fields, including transportation, surveying, geodesy, and geophysics. However, the ionosphere, troposphere, and multi-path effects can lead to measurement errors, including the so-called integer ambiguity, which is one of the most challenging issues in high-precision GNSS applications.

The integer ambiguity problem arises when the carrier-phase measurements from GNSS signals are not uniquely determined, leading to an integer number of cycles that need to be resolved in order to achieve high-precision positioning. Lattice theory has been widely used in the field of integer ambiguity resolution, and lattice reduction algorithms such as LLL have been used to reduce the correlation between lattice bases and achieve better resolution results.

However, the traditional LLL algorithms have limitations in terms of computational efficiency and solution quality. To address these issues, we propose an improved deep-insertion LLL algorithm (DLLL) for integer ambiguity resolution. The DLLL algorithm is a variant of the deep-insertion LLL algorithm, which can effectively reduce the correlation between lattice bases while also achieving better solution quality and computational efficiency than traditional LLL algorithms.

The DLLL algorithm has several key advantages over traditional LLL algorithms. First, it can reduce the correlation between lattice bases more effectively, which leads to better resolution results. Second, it can achieve better solution quality and computational efficiency, which are crucial for high-precision GNSS applications. Third, it can handle large-scale integer ambiguity problems more efficiently than other state-of-the-art algorithms.

To evaluate the performance of the DLLL algorithm, we conduct simulation experiments and compare it with traditional LLL algorithms and other state-of-the-art algorithms. The results show that the DLLL algorithm can significantly improve the resolution quality and computational efficiency, making it a promising approach for integer ambiguity resolution in high-precision GNSS applications.

Furthermore, we investigate the impact of the choice of lattice basis and the number of epochs on the performance of the DLLL algorithm. Our results suggest that selecting a suitable lattice basis and increasing the number of epochs can further improve the resolution quality and computational efficiency of the DLLL algorithm.

In conclusion, the proposed DLLL algorithm provides a new solution for the challenging problem of integer ambiguity resolution in high-precision GNSS applications. Its effectiveness and efficiency make it a promising approach for future research and practical applications in this field. The improvement of this algorithm can greatly enhance the accuracy of positioning and navigation, and has significant implications for various areas such as geodesy, surveying, transportation, and geophysics.

Moreover, the DLLL algorithm has potential applications beyond GNSS. For example, it can be used in the field of wireless communications to solve similar problems of phase ambiguity in signal processing. The algorithm can also be extended to other areas such as cryptography, coding theory, and optimization, where lattice reduction algorithms have been widely used.

Finally, future research can focus on further improving the efficiency and scalability of the DLLL algorithm, as well as exploring its applications in other areas. The algorithm can be combined with other techniques such as machine learning and artificial intelligence to achieve even better results. The development of advanced algorithms for integer ambiguity resolution has the potential to revolutionize the field of high-precision positioning and navigation, and to bring significant benefits to society.