Regular Simplices Inscribed into the Cube and Exhibiting a Group Structure

Abstract

For $n\in \mathbb{N}$, we interpret the vertex set $W_n$ of the $n$-cube as a vector space over the field ${\mathbb{F}}_2$ and prove that a regular $n$-simplex can be inscribed into the $n$-cube such that its vertices constitute a subgroup of $W_n$ if and only if $n+1$ is a power of 2. Furthermore, a connection to the theory of Hamming Codes will be established.