Facets in the Vietoris--Rips complexes of hypercubes

In this paper, we investigate the facets of the Vietoris--Rips complex $\mathcal{VR}(Q_n; r)$ where $Q_n$ denotes the $n$-dimensional hypercube. We are particularly interested in those facets which are somehow independent of the dimension $n$. Using Hadamard matrices, we prove that the number of different dimensions of such facets is a super-polynomial function of the scale $r$, assuming that $n$ is sufficiently large. We show also that the $(2r-1)$-th dimensional homology of the complex $\mathcal{VR}(Q_n; r)$ is non-trivial when $n$ is large enough, provided that the Hadamard matrix of order $2r$ exists.