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.