The use of Global Navigation Satellite Systems (GNSS) has become ubiquitous in modern society, enabling accurate and reliable positioning and navigation in a wide range of applications. GNSS positioning and navigation rely on the estimation of the integer number of cycles completed by the GNSS signal, which is known as the integer ambiguity. This estimation process is challenging due to various sources of noise and error in the measurements, which can cause incorrect or biased ambiguity estimates.

In recent years, lattice theory has emerged as a powerful tool for integer ambiguity resolution in GNSS. Lattice theory provides a mathematical framework for representing and analyzing the ambiguity problem in GNSS. In this framework, the GNSS measurements are represented as a lattice in a high-dimensional space, with the lattice basis vectors representing the phase observations. The goal is to find a set of integer coefficients that correspond to the true phase values.

To improve the efficiency and accuracy of the solution, reducing the correlation between lattice bases is crucial. This can be achieved through the process of lattice reduction, which involves both scale reduction and basis vector exchange. The Lenstra Lenstra Lovász (LLL) algorithm has been widely used for lattice reduction in GNSS. Still, to further enhance its efficiency, the Deep LLL (DLLL) method was developed. Building on this, a new algorithm called the Improved Deep LLL (IDLLL) method has been proposed, which utilizes a sorting matrix and modified column norms to reduce the number of column norm calculations required.

Experimental results have shown that the IDLLL algorithm outperforms both LLL and DLLL reduction algorithms in terms of reduction efficiency and effectiveness. The theoretical analysis has also supported the superiority of the IDLLL algorithm. The use of the IDLLL algorithm in GNSS integer ambiguity resolution has significant potential to improve accuracy and reliability.

However, the use of lattice theory in GNSS ambiguity resolution also presents challenges. The high dimensionality of the lattice space and the complexity of the GNSS measurements can lead to significant computational demands. Additionally, the performance of the lattice reduction algorithms can be sensitive to the choice of parameters and the specific characteristics of the GNSS signals.

Overall, the application of lattice theory in GNSS ambiguity resolution is an exciting and promising field of research. The development of the IDLLL algorithm is a significant step forward in improving the efficiency and accuracy of the solution. Further research is needed to explore the full potential of lattice theory in GNSS positioning and navigation applications, including the development of more efficient algorithms and the investigation of other applications of lattice theory in GNSS.