$(\sigma ,\delta )$-polycyclic codes in Ore extensions over rings

Abstract

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.

20 pages, no figure