Moduli Selection in Robust Chinese Remainder Theorem: Closed-Form Solutions and Layered Design

We study the fundamental problem of \emph{moduli selection} in the Robust Chinese Remainder Theorem (RCRT), where each residue may be perturbed by a bounded error. Consider $L$ moduli of the form $m_i = \Gamma_i m$ ($1 \le i \le L$), where $\Gamma_i$ are pairwise coprime integers and $m \in \mathbb{R}^+$ is a common scaling factor. For small $L$ ($L = 2, 3, 4$), we obtain exact solutions that maximize the robustness margin under dynamic-range and modulus-bound constraints. We also introduce a Fibonacci-inspired \emph{layered} construction (for $L = 2$) that produces exactly $K$ robust decoding layers, enabling predictable trade-offs between error tolerance and dynamic range. We further analyze how robustness and range evolve across layers and provide a closed-form expression to estimate the success probability under common data and noise models. The results are promising for various applications, such as sub-Nyquist sampling, phase unwrapping, range estimation, modulo analog-to-digital converters (ADCs), and robust residue-number-system (RNS)-based accelerators for deep learning. Our framework thus establishes a general theory of moduli design for RCRT, complementing prior algorithmic work and underscoring the broad relevance of robust moduli design across diverse information-processing domains.