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 $\mathscr{R}$.

翻译：本文研究了定义在特定串行环类上的多循环码的代数结构，该环定义为 $\mathscr R=R[x_1,\ldots, x_s]/\langle t_1(x_1),\ldots, t_s(x_s) \rangle$，其中 $R$ 为链环，且对每个 $i\in\{1,\ldots, s\}$，$t_i(x_i)$ 是 $R[x_i]$ 中的首一平方自由多项式。我们定义了拟 $s$ 维多循环码，并建立了这些码与 $\mathscr R$ 上多循环码之间的 $R$-同构。我们给出了在这类环上存在零化子自对偶、零化子自正交、零化子线性互补对偶以及零化子对偶包含多循环码的充分必要条件。我们还建立了链环 $R$ 上零化子对偶保持多循环码的CSS构造，并利用该构造从 $\mathscr{R}$ 上的多循环码导出了量子码。