On duals of multiplicative designs

A multiplicative design is a square design (that is, a set S of n elements called varieties, and a collection of n subsets of S called blocks) in which each block may be assigned a positive number, called the block's weight, less than the size of the block in such a way that the size of the intersection of two distinct blocks is the geometric mean of their weights. A uniform design is a multiplicative design in which the difference between the weight and size of a block is independent of the choice of the block. A λ-design is a multiplicative design with identical weights in which not all of the block sizes are equal. It is conjectured that if a multiplicative design has a multiplicative dual, and if neither design belongs to a specific class of designs, then both are uniform designs. Two cases of this conjecture are proved, one of which is this generalization of a result of K. N. Majumdar: a λ-design with a multiplicative dual has λ = 1. Degenerate multiplicative designs are investigated. A generalization to multiplicative designs of Henry B. Mann's upper bound on the multiplicity of a repeated variety is also proved.