On the structure of $1$-generator quasi-polycyclic codes over finite chain rings

Abstract

Quasi-polycyclic (QP for short) codes over a finite chain ring $R$ are a generalization of quasi-cyclic codes, and these codes can be viewed as an $R[x]$-submodule of ${\mathcal{R}}_m^{\ell }$, where ${\mathcal{R}}_m:=R[x]/⟨f⟩$, and $f$ is a monic polynomial of degree $m$ over $R$. If $f$ factors uniquely into monic and coprime basic irreducibles, then their algebraic structure allow us to characterize the generator polynomials and the minimal generating sets of 1-generator QP codes as $R$-modules. In addition, we also determine the parity check polynomials for these codes by using the strong Gröbner bases. In particular, via Magma system, some quaternary codes with new parameters are derived from these 1-generator QP codes.