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 $\mathcal{R}_m:= R[x]/\langle f\rangle$, 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 Gr\"{o}bner bases. In particular, via Magma system, some quaternary codes with new parameters are derived from these 1-generator QP codes.

翻译：用于固定链环的微粒周期(QP)代码 $R$是准周期代码的概括,这些代码可被视为$[x]$的子模块,其中$=mathcal{R ⁇ m:=R[x]/\langle f\rangle $;$f$是一元多元度,单位超过$R$。如果美元是单子和共性基本代码中独有的硬度和共性要素,那么它们的代谢结构允许我们将生成的多元代码和最小生成的一元QP代码定性为$R$。此外,我们还通过使用坚固的 Gr\"{o}bner基底来确定这些代码的等值检查多元值。特别是,通过Magma系统,一些带有新参数的四元代码来自这些1-gener QP 代码。