Rectangles, integer vectors and hyperplanes of the hypercube

Abstract

We introduce a family of nonnegative integer vectors - primitive vectors - defining hyperplanes of the real affine cube over $C^n:=\{-1,1{\}}^n$ and study their properties with respect to the rectangles of the cube. As a consequence we give a short proof that, for small dimensions ($n\le 7$), the real affine cube can be recovered from its signed rectangles and its signed cocircuits complementary of its facets and skew-facets.