Balanced bipartite weighing designs

A block B denotes a set of k = k₁ + k₂ elements which are divided into two subsets B¹ and B², where |Bⁱ| = k_i, i = 1 or 2. Two elements of B are said to be linked or n-linked in B if and only if they belong to different subsets or the same subset of B respectively. A balanced bipartite weighing design, (briefly BBWD (υ, k₁, k₂, λ₁)) is an arrangement of υ elements into b blocks, each containing k elements, such that each element occurs in exactly r blocks, any two distinct elements are linked in exactly λ₁ blocks and n-linked in exactly λ₂ blocks.Given fixed k₁ and k₂, there is always a minimal value of λ₁ such that the necessary conditions for the existence of a BBWD are satisfied for same υ. It is proved that in many cases, the necessary conditions are also sufficient. Some general methods for constructing BBWD's as well as a table of all designs with υ ? 13 are obtained.