Characterizing imprimitive partition designs of binary Hamming graphs (Q1827338)

Let \(m\in N\). The binary Hamming graph of diameter \(m\), \(H(m)\), has as vertices the elements of \(F^m\). The edges join vertices at distance one, where the distance between two vertices is the number of different coordinates. Since the Hamming graph \(H(m)\) is distance-regular, the distance partition from any vertex is a partition design. To find all partition designs that are characterized by their adjacency matrix seems to be a very difficult problem. Instead of considering all partition designs, only imprimitive partition designs have been considered in this paper. Let \(G= (V,E)\) be a binary Hamming graph (or 1-skeleton of a hypercube). A partition design of \(G\) with adjacency matrix \(M= (m_{ij})_{1\leq i,j\leq r}\) is defined as a partition \(\{Y_1,Y_2,\dots, Y_r\}\) of the vertex set \(V\) such that for every \(x\in Y_i\) we can have \(|\{y\in Y_j\mid (x,y)\in E\}|= m_{ij}\); this holds for \(1\leq i,\,j\leq r\). Let \(Y\) be a partition design with adjacency matrix \(M\). For every \(t\geq 2\), the author constructs a partition design \(Y^t\) with adjacency matrix \(tM\), and describes when \(Y^t\) is the unique partition design with adjacency matrix \(tM\).