Systematic sampling is a minimal support design

In order to select a sample in a finite population of N units with given inclusion probabilities, it is possible to define asamplingdesign on at most N samples that have a positive probability of being selected. Designs defined on minimal sets of samples are called minimum supportdesigns. It is shown that, for any vector of inclusion probabilities, systematicsampling always provides a minimum supportdesign. This property makes it possible to extensively compute the samplingdesign and the joint inclusion probabilities. Random systematicsampling can be viewed as the random choice of a minimum supportdesign. However, even if the population is randomly sorted, a simple example shows that some joint inclusion probabilities can be equal to zero. Another way of randomly selecting a minimum supportdesign is proposed, in such a way that all the samples have a positive probability of being selected, and all the joint inclusion probabilities are positive.