  In this paper, we consider the orthogonal product set (OPS) with strong
quantum nonlocality. Based on the decomposition of plane geometry, we present a
sufficient condition for the triviality of orthogonality-preserving POVM on
fixed subsystem and partially answer an open question given by Yuan et al.
Phys. Rev. A \textbf{102}, 042228 (2020)}. The connection between the
nonlocality and the plane structure of OPS is established. We successfully
construct a strongly nonlocal OPS in $\mathcal{C}^{d_{A}}\otimes
\mathcal{C}^{d_{B}}\otimes \mathcal{C}^{d_{C}}$ $(d_{A,B,C}\geq 4)$, which
contains fewer quantum states, and generalize the structures of known OPSs to
any possible three and four-partite systems. In addition, we present several
entanglement-assisted protocols for perfectly local discrimination the sets. It
is shown that the protocols without teleportation use less entanglement
resources on average and these sets can always be discriminated locally with
multiple copies of 2-qubit maximally entangled states. These results also
exhibit nontrivial signification of maximally entangled states in the local
discrimination of quantum states.
