Cubes of symmetric designs

Abstract

We study n-dimensional matrices with {0, 1}-entries (n-cubes) such that all their 2-dimensional slices are incidence matrices of symmetric designs. A known construction of these objects obtained from difference sets is generalized so that the resulting n-cubes may have inequivalent slices. For suitable parameters, they can be transformed into ndimensional Hadamard matrices with this property. In contrast, previously known constructions of n-dimensional designs all give examples with equivalent slices.