An example related to the best code of reconstructing the cells of a partition design from its adjacency matrix

We describe how eight copies of the Best code can be imbedded in the binary Hamming graph of length 10 in such a way that they form part of a partition design π1. We study those isometries of the binary Hamming graph which stabilize π1. A partition design π2, coarser than π1, will be the subject of the second part of this work. Let M be the adjacency matrix of π2. We will show that M is extreme in some sense. Namely, π2 has the following interesting property: π2 is the unique partition design with adjacency matrix M.