Polycyclic codes over serial rings and their annihilator CSS construction

Abstract

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

24 pages, version accepted for publication in Cryptography and Communications