Projection cubes of symmetric designs

We introduce a new type of $n$-dimensional generalization of symmetric $(v,k,\lambda)$ block designs. We prove an upper bound on the dimension $n$ in terms of $v$. We also define the corresponding concept of $n$-dimensional difference sets and extend some classic families of difference sets to higher dimensions. Complete classifications are performed for small parameters $(v,k,\lambda)$ and some interesting examples are presented.