  Recently, Zhang et al [Phys. Rev. A 92, 012332 (2015)] presented $4d-4$
orthogonal product states that are locally indistinguishable and completable in
a $d\otimes d$ quantum system. Later, Zhang et al. [arXiv: 1509.01814v2 (2015)]
constructed $2n-1$ orthogonal product states that are locally indistinguishable
in $m\otimes n$ ($3\leq m \leq n$). In this paper, we construct a locally
indistinguishable and completable orthogonal product basis with $4p-4$ members
in a general $m\otimes n$ ($3\leq m \leq n$) quantum system, where $p$ is an
arbitrary integer from $3$ to $m$, and give a very simple but quite effective
proof for its local indistinguishability. Specially, we get a completable
orthogonal product basis with $8$ members that cannot be locally distinguished
in $m\otimes n$ ($3\leq m \leq n$) when $p=3$. It is so far the smallest
completable orthogonal product basis that cannot be locally distinguished in a
$m\otimes n$ quantum system. On the other hand, we construct a small locally
indistinguishable orthogonal product basis with $2p-1$ members, which is maybe
uncompletable, in $m\otimes n$ ($3\leq m \leq n$ and $p$ is an arbitrary
integer from $3$ to $m$). We also prove its local indistinguishability. As a
corollary, we give an uncompletable orthogonal product basis with $5$ members
that are locally indistinguishable in $m\otimes n$ ($3\leq m \leq n$). All the
results can lead us to a better understanding of the structure of a locally
indistinguishable product basis in $m \otimes n$.
