  A set of orthogonal product states is strongly nonlocal if it is locally
irreducible in every bipartition, which shows the phenomenon of strong quantum
nonlocality without entanglement. Although such a phenomenon has been shown to
any three-, four-, and five-partite systems, the existence of strongly nonlocal
orthogonal product sets in multipartite systems remains unknown. In this paper,
by using a general decomposition of the $N$-dimensional hypercubes, we present
strongly nonlocal orthogonal product sets in $N$-partite systems for all odd
$N\geq 3$. Based on this decomposition, we give explicit constructions of
unextendible product bases in $N$-partite systems for odd $N\geq 3$.
Furthermore, we apply our results to quantum secret sharing, uncompletable
product bases, and PPT entangled states.
