In this paper, we study the algebraic structure of $(\sigma,\delta)$-polycyclic codes as submodules in the quotient module $S/Sf$, where $S=R[x,\sigma,\delta]$ is the Ore extension, $f\in S$, and $R$ is a finite but not necessarily commutative ring. We establish that the Euclidean duals of $(\sigma,\delta)$-polycyclic codes are $(\sigma,\delta)$-sequential codes. By using $(\sigma,\delta)$-Pseudo Linear Transformation (PLT), we define the annihilator dual of $(\sigma,\delta)$-polycyclic codes. Then, we demonstrate that the annihilator duals of $(\sigma,\delta)$-polycyclic codes maintain their $(\sigma,\delta)$-polycyclic nature. Furthermore, we classify when two $(\sigma,\delta)$-polycyclic codes are Hamming isometrical equivalent. By employing Wedderburn polynomials, we introduce simple-root $(\sigma,\delta)$-polycyclic codes. Subsequently, we define the $(\sigma, \delta)$-Mattson-Solomon transform for this class of codes and we address the problem of decomposing these codes by using the properties of Wedderburn polynomials.

翻译：本文研究了Ore扩张$S=R[x,\sigma,\delta]$中$(\sigma,\delta)$-多循环码作为商模$S/Sf$子模的代数结构，其中$f\in S$，$R$为有限但未必交换的环。我们证明了$(\sigma,\delta)$-多循环码的欧几里得对偶是$(\sigma,\delta)$-序列码。通过使用$(\sigma,\delta)$-伪线性变换(PLT)，定义了$(\sigma,\delta)$-多循环码的零化子对偶，并证明零化子对偶保持$(\sigma,\delta)$-多循环性质。进一步地，我们分类了两类$(\sigma,\delta)$-多循环码在Hamming等距意义下等价的充要条件。借助Wedderburn多项式，引入了单根$(\sigma,\delta)$-多循环码，并定义了该类码的$(\sigma,\delta)$-Mattson-Solomon变换，最后利用Wedderburn多项式的性质研究了码的分解问题。