Polycyclic codes over serial rings and their annihilator CSS construction

In this paper, we investigate the algebraic structure for polycyclic codes over a specific class of serial rings, defined as \(\mathscr {R}=R[x_1,\ldots , x_s]/\langle t_1(x_1),\ldots , t_s(x_s) \rangle \) $\mathcal{R}=R[x_1,\dots ,x_s]/⟨t_1(x_1),\dots ,t_s(x_s)⟩$ , where R is a chain ring and each \(t_i(x_i)\) $t_i(x_i)$ in \(R[x_i]\) $R[x_i]$ for \(i\in \{1,\ldots , s\}\) $i\in \{1,\dots ,s\}$ 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 \(\mathscr {R}\) $\mathcal{R}$ . 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 \(\mathscr {R}\) $\mathcal{R}$ .