Low-Density Parity-Check (LDPC) codes have low linear decoding complexity, which is a kind of good codes with excellent performance. Therefore, LDPC codes have great research value. Two kinds of LDPC codes are constructed based on vector space over finite field. The code length, code rate and minimum distance are given. Moreover, the two kinds of codes are compared with the existing codes, and the constructed codes are better than some existing ones in terms of code rate or minimum distance.

Compressed Sensing (CS) is a new signal processing theory under the condition that the signal is sparse or compressible. One of the central problems in compressed sensing is the construction of sensing matrices. In this paper, we provide a new deterministic construction via vector spaces over finite fields, which is superior to Devore's construction using polynomials over finite fields under some conditions. Moreover, we use the algorithm to perform numerical simulation experiments on sensing matrices. Simulation results also demonstrate that signal recovery performance performs better using the constructed matrices as compared with several state-of-the-art sensing matrices, such as DeVore's matrix and random Gaussian matrix.