Construction of two \((15, 3,1)\) block designs (Q1913325)

It is known that there are 80 distinct 2--(15,3,1) designs, of which the classical one is the points and lines of \(PG (3,2)\). Here, two non-isomorphic designs are constructed from the \(10_3\) configuration whose points are the vertices of a tetrahedron \(T\) and the midpoints of its edges in an affine space over a field of characteristic not 2 or 3. Adding the centroids of the faces and the centroid of \(T\) completes the 15 points. The two designs are constructed from these 15 points, and one is \(PG (3,2)\).