  In this paper, we investigate the algebraic structure for polycyclic codes
over a specific class of serial rings, defined as $\mathscr R=R[x_1,\ldots,
x_s]/\langle t_1(x_1),\ldots, t_s(x_s) \rangle$, where $R$ is a chain ring and
each $t_i(x_i)$ in $R[x_i]$ for $i\in\{1,\ldots, s\}$ is a monic square-free
polynomial. We define quasi-$s$-dimensional polycyclic codes and establish an
$R$-isomorphism between these codes and polycyclic codes over $\mathscr R$. We
provide necessary and sufficient conditions for the existence of annihilator
self-dual, annihilator self-orthogonal, annihilator linear complementary dual,
and annihilator dual-containing polycyclic codes over this class of rings. We
also establish the CSS construction for annihilator dual-preserving polycyclic
codes over the chain ring $R$ and use this construction to derive quantum codes
from polycyclic codes over $\mathscr{R}$.
