In this paper, we study the algebraic structure of $(\sigma,\delta)$-polycyclic codes, defined as submodules in the quotient module $S/Sf$, where $S=R[x,\sigma,\delta]$ is the Ore extension ring, $f\in S$, 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, 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$(σ,δ)$-多循环码的代数结构，将其定义为商模$S/Sf$中的子模，其中$S=R[x,σ,δ]$为Ore扩张环，$f∈S$，而$R$为有限但未必交换的环。我们证明了$(σ,δ)$-多循环码的欧几里得对偶是$(σ,δ)$-序列码。通过利用$(σ,δ)$-伪线性变换，定义了$(σ,δ)$-多循环码的零化子对偶，并证明了该对偶仍保持$(σ,δ)$-多循环性质。此外，我们分类了两个$(σ,δ)$-多循环码具有汉明等距等价的条件。借助Wedderburn多项式，引入了单根$(σ,δ)$-多循环码，进而定义了这类码的$(σ,δ)$-马特森-所罗门变换，并利用Wedderburn多项式的性质解决了此类码的分解问题。