Symmetric (15, 8, 4) -designs in terms of the geometry of binary simplex codes of dimension 4

Let n=2ᵏ-1 and m=2^k-2 for a certain k 3. Consider the point-line geometry of 2m-element subsets of an n-element set. Maximal singular subspaces of this geometry correspond to binary simplex codes of dimension k. For k 4 the associated collinearity graph contains maximal cliques different from maximal singular subspaces. We investigate maximal cliques corresponding to symmetric (n, 2m, m) -designs. The main results concern the case k=4 and give a geometric interpretation of the five well-known symmetric (15, 8, 4) -designs.