Optimal design with many blocking factors

Designs for sets of experimental units with many blocking factors are studi ed. It is shown that if the set of blocking factors satisfies a certain sim ple condition then the information matrix for the design has a simple form. In consequence, a design is optimal if it is optimal with respect to one p articular blocking factor and regular with respect to all the rest, in a se nse which is made precise in the paper. This encompasses several previous r esults for optimal designs with more than one blocking factor, and applicat ions to many other situations are given.