The construction of deterministic sensing matrices satisfying the Restricted Isometry Property (RIP) remains a fundamental challenge in compressed sensing. This paper introduces a novel deterministic framework for constructing sensing matrices by exploiting the algebraic structure of the lattice of integers under the divisibility partial order. We demonstrate that the Möbius function and incidence algebra associated with this lattice naturally give rise to matrices with low coherence and structured sparsity. The proposed construction yields matrices of size φ(N) × N, where φφis Euler\\\'s totient function, with column coherence bounded by O (1/√(φ(N))), asymptotically achieving the Welch bound. Unlike random constructions, our approach guarantees perfect recovery for signals sparse in the standard basis without probabilistic arguments. Furthermore, we establish a connection between the divisibility lattice and Dirichlet convolution, enabling efficient signal reconstruction via number-theoretic transforms. Numerical experiments validate the theoretical guarantees and demonstrate competitive performance against existing deterministic constructions based on finite fields and algebraic geometry.
Anuradha et al. (Thu,) studied this question.