Projection cubes of symmetric designs

We introduce a new type of $n$-dimensional generalization of symmetric $(v,k,\lambda )$ block designs. We prove upper bounds on the dimension $n$ in terms of $v$ and $k$. 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.