Abstract Nowadays, the deterministic construction of sensing matrices is a hot topic in compressed sensing. The coherence of the measurement matrix is an important research area in the design of deterministic compressed sensing. To solve this problem, this paper proposes a novel sparse deterministic measurement matrix, which basically accesses the optimal low coherence of the measurement matrix. Firstly, a class of sparse square matrix is constructed based on finite fields' arithmetic. Then, the Hadamard matrix, or (discrete Fourier transform) DFT matrix, is nestled into the square matrix to construct an asymptotically optimal deterministic measurement matrix. That is, the relevant column vectors have orthogonal characteristics. Using this feature, the measurement matrix can be further optimized to reduce its mutual coherence, almost achieving the lower bound of the coherence (Welch bound). The two types of deterministic measurement matrices proposed are sparse with low mutual coherence and flexible measurement sizes. So, the proposed deterministic measurement matrices require less memory and time for the recovery as well as reducing the complexity due to their sparse structure. The simulation results show that compared with the existing (several typical) random matrices, the proposed method can reduce the mutual coherence and computational complexity of the measurement matrix.