All hyperbolic Coxeter $n$-cubes

Beside simplices, $n$-cubes form an important class of simple polyhedra. Unlike hyperbolic Coxeter simplices, hyperbolic Coxeter $n$-cubes are not classified. We show that there is no hyperbolic Coxeter $n$-cube for $n\geq~6$, and provide a full classification for $n\leq 5$. Our methods, which are essentially of combinatorial and algebraic nature, can be (and have been successfully) implemented in a symbolic computation software such as Mathematica$^\circledR$.