A Composition Theorem for Simple Designs and Difference Families

A method is presented for constructing simple partially balanced designs from \(t-(v,k,\lambda)\) designs. When the component designs satisfy a compatibility condition the result is a simple balanced design. The component designs can even be trivial (with some exceptions) with the resulting design being nontrivial. The automorphism group of the composition is given in terms of the automorphism groups of the component designs. Some previously unknown simple designs are constructed, including an infinite family of \(3\)-designs that are extremal with respect to an inequality of Cameron and Praeger. Some analogous theorems are given for difference families.