The development of vaccines for COVID-19 has enabled us to nearly return to pre-pandemic life. However, while vaccines are becoming globally widespread, high alert levels prevail. Even with vaccines, monitoring for the evolution of mutations or detecting new outbreaks calls for continued vigilance. Therefore, testing is likely to prevail to be a vital mechanism to inform decision making in the near future. In order to conserve scarce testing resources, many nations have endorsed so-called group/pooling test methods. Such methods can be expressed using linear algebra. The basic principle underlying pooling tests is the observation that to efficiently detect positive cases among a population with a very low occurrence prevalence, it can be advantageous to test groups of samples instead of testing all individual samples. We develop matrix designs, which encode all relevant information for doing pooling tests and that enable high compression rates when exactly identifying up to a certain number of positive cases.