Doubly equivalent designs

Mendelsohn and Liang, in J. Combin. Des. 11 (2003), introduced a new class of difference sets and designs: (v, k, [λ1, λ2, . . . , λm])-difference sets and (v, k, [λ1, λ2, . . . , λm])-designs, and mainly discussed designs with a relationship called λ-equivalence. In this paper, we introduce the concepts of a doubly equivalent design as well as a super class. We describe the structure of super classes and discuss properties of doubly equivalent designs. We generalize results of simply equivalent designs in the paper of Mendelsohn and Liang to doubly equivalent designs. The main result is Theorem 3.7, which claims that a particular class of doubly equivalent designs can produce singly equivalent designs.