  Quasi-polycyclic (QP for short) codes over a finite chain ring $R$ are a
generalization of quasi-cyclic codes, and these codes can be viewed as an
$R[x]$-submodule of $\mathcal{R}_m^{\ell}$, where $\mathcal{R}_m:= R[x]/\langle
f\rangle$, and $f$ is a monic polynomial of degree $m$ over $R$. If $f$ factors
uniquely into monic and coprime basic irreducibles, then their algebraic
structure allow us to characterize the generator polynomials and the minimal
generating sets of 1-generator QP codes as $R$-modules. In addition, we also
determine the parity check polynomials for these codes by using the strong
Gr\"{o}bner bases. In particular, via Magma system, some quaternary codes with
new parameters are derived from these 1-generator QP codes.
