Rectangles, integer vectors and hyperplanes of the hypercube

TL;DR

This paper studies primitive integer vectors defining hyperplanes in the hypercube and shows how the hypercube can be reconstructed from certain signed rectangles and cocircuits for dimensions up to 7.

Contribution

It introduces primitive vectors related to hyperplanes of the hypercube and proves a reconstruction result for the hypercube from signed rectangles and cocircuits in small dimensions.

Findings

Hyperplanes of the hypercube can be characterized by primitive vectors.

The hypercube is reconstructible from signed rectangles and cocircuits for n ≤ 7.

Provides a new proof for the reconstruction property in low dimensions.

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 ($n\leq 7$), the real affine cube can be recovered from its signed rectangles and its signed cocircuits complementary of its facets and skew-facets.