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 R=R[x₁,…, xₛ]/ t₁(x₁),…, tₛ(xₛ), where R is a chain ring and each tᵢ(xᵢ) in R[xᵢ] for i∈\1,…, 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 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 R.