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

Let n=2ᵏ-1 and m=2ᵏ⁻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.