Sidon space is an important tool for constructing cyclic subspace codes. In this letter, we construct some Sidon spaces by using primitive elements and the roots of some irreducible polynomials over finite fields. Let q be a prime power, k, m, n be three positive integers and $\rho= \lceil \frac{m}{2k}\rceil-1$, $\theta= \lceil \frac{n}{2m}\rceil-1$. Based on these Sidon spaces and the union of some Sidon spaces, new cyclic subspace codes with size $\frac{3(q^{n}-1)}{q-1}$ and $\frac{\theta\rho q^{k}(q^{n}-1)}{q-1}$ are obtained. The size of these codes is lager compared to the known constructions from [14] and [10].

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.

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.

Faced with a large number of samples to be tested, if there are requiring to be tested one by one and complete in a short time, it is difficult to save time and save costs at the same time. The random pooling designs can deal with it to some degree. In this paper, a family of random pooling designs based on the singular linear spaces and related counting theorems are constructed. Furtherly, based on it we construct an $ \alpha $-$ almost\ d^e $-disjunct matrix and an $ \alpha $-$ almost\ (d, r, z] $-disjunct matrix, and all the parameters and properties of these random pooling designs are given. At last, by comparing to Li's construction, we find that our design is better under certain condition.