Generalized Multidimensional Chinese Remainder Theorem (MD-CRT) for Multiple Integer Vectors

Chinese Remainder Theorem (CRT) has wide applications in cryptography, coding theory, and signal processing. Its multidimensional extension (MD-CRT) reconstructs an integer vector from its vector remainders modulo several integer matrices. This paper studies a generalized MD-CRT for multiple integer vectors, where the objective is to reconstruct several integer vectors from multiple vector residue sets modulo multiple integer matrices. Unlike the scalar case, i.e., the generalized CRT, the moduli here are matrices that do not commute, and the uniquely determinable range becomes multidimensional with more complex inclusion relations. We address two fundamental questions. First, we derive a uniquely determinable range without prior information and propose a corresponding reconstruction algorithm. Second, we investigate the case of two integer vectors and establish a new condition for achieving the maximal dynamic range. Notably, when reduced to one dimension, this condition is strictly weaker than all the previously known ones. These results may have applications for frequency detection in multidimensional signal processing.