Partition Designs with two subsets of the Hamming Space

A partition design with two sets in the binary Hamming graph has associated its (2×2 positive integer) adjacency matrix. We consider the problem of finding all such matrices. We will prove that the matrix abcd is of this type if and only if d=a + b-c, (ii)b+cgcd(b,c) is a power of two; and (iii) a ⩾ f(b, c), where f is an unknown function but with the following bounds:12−(b−c)+ 1/4 + c(b−c) ⩽ f(b,c) ⩽ c − gcd(b, c) for b ⩾ c. l also treat the problem of determining if a matrix is the adjacency matrix of a unique (up to isometry) partition design of the binary Hamming graph.