Yuri Evangelista, Alessio Nuti, Francesco Ceraudo, Edoardo Giancarli, Giuseppe Dilillo, Riccardo Campana, Giovanni Della Casa, Ettore Del Monte, Marco Feroci, Mauro Fiorini, Giovanni Lombardi, Massimo Rapisarda, Francesca Esposito, Immacolata Donnarumma, Alessandro Turchi, Ugo Cortesi, Fabio D'Amico, Marco Gai, Andrea Argan

The Lunar Electromagnetic Monitor in X-rays (LEM-X) is a proposed wide-field X-ray observatory designed for deployment on the Moon's surface. Its primary scientific goal is to enhance multi-messenger astrophysics by detecting, localizing, and monitoring high-energy transient phenomena and variable X-ray sources across the sky. Building on the heritage of the eXTP and LOFT mission proposals, LEM-X employs pairs of coded-aperture cameras equipped with large-area linear Silicon Drift Detectors (SDDs), offering excellent spectral resolution ($\leq$350 eV at 6 keV) over the 2-50 keV energy range. Each camera provides a field of view of ~1 steradian at 25% effective area and achieves a Point-Source Location Accuracy (PSLA) of 1 arcminute, with an on-axis sensitivity better than 5 mCrab in 50 ks and 700 mCrab in 1 s. In this paper we describe the experiment and focus on the detailed design and optimization of the LEM-X coded mask, analyzing its scientific performance, imaging capabilities, and thermo-mechanical properties. We describe the mask code generation, decoding algorithms, and the trade-offs involved in achieving the required angular resolution, sensitivity, and structural integrity. Imaging simulations and mechanical analyses confirm the effectiveness of the proposed design, demonstrating its suitability for high-precision, wide-field X-ray imaging devoted to multi-messenger astrophysics and transient events detection.