Characterizing imprimitive partition designs of binary Hamming graphs

Let G=(V,E) be a binary Hamming graph (or the 1-skeleton of a hypercube). A partition design of G with adjacency matrix M=(mij)1≤i,j≤r is defined as a partition {Y1,…,Yr} of the vertex set V such that for every x∈Yi we have that |{y∈Yj∣(x,y)∈E}|=mij; this holds for 1≤i,j≤r. be a partition design with adjacency matrix M. For every t≥2 we construct a partition design Yt with adjacency matrix tM, and we describe when Yt is the unique partition design with adjacency matrix tM.