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A set of orthogonal product states is strongly nonlocal if it is locallyirreducible in every bipartition, which shows the phenomenon of strong quantumnonlocality without entanglement. Although such phenomenon has been shown toany three-, four-, and five-partite systems, the existence of strongly nonlocalorthogonal product sets in multipartite systems remains unknown. In this paper,by using a general decomposition of the $N$-dimensional hypercubes for odd$N\geq 3$, we show strongly nonlocal orthogonal product sets in $N$-partitesystems for odd $N\geq 3$. Moreover, basing on this decomposition, we giveexplicit construction of unextendible product bases in $N$-partite systems forodd $N\geq 3$.

Strong quantum nonlocality and unextendibility without entanglement in $N$-partite systems with odd $N$