$(σ,δ)$-polycyclic codes in Ore extensions over rings
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arXiv
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| Autori principali: | , , , |
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| Natura: | Preprint |
| Pubblicazione: |
2023
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| _version_ | 1866916141924876288 |
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| author | Bajalan, Maryam Landjev, Ivan Martínez-Moro, Edgar Szabo, Steve |
| author_facet | Bajalan, Maryam Landjev, Ivan Martínez-Moro, Edgar Szabo, Steve |
| contents | In this paper, we study the algebraic structure of $(σ,δ)$-polycyclic codes, defined as submodules in the quotient module $S/Sf$, where $S=R[x,σ,δ]$ 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 $(σ,δ)$-polycyclic codes are $(σ,δ)$-sequential codes. By using $(σ,δ)$-Pseudo Linear Transformation, we define the annihilator dual of $(σ,δ)$-polycyclic codes. Then, we demonstrate that the annihilator duals of $(σ,δ)$-polycyclic codes maintain their $(σ,δ)$-polycyclic nature. Furthermore, we classify when two $(σ,δ)$-polycyclic codes are Hamming isometrical equivalent. By employing Wedderburn polynomials, we introduce simple-root $(σ,δ)$-polycyclic codes. Subsequently, we define the $(σ, δ)$-Mattson-Solomon transform for this class of codes and we address the problem of decomposing these codes by using the properties of Wedderburn polynomials. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2312_07193 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | $(σ,δ)$-polycyclic codes in Ore extensions over rings Bajalan, Maryam Landjev, Ivan Martínez-Moro, Edgar Szabo, Steve Information Theory In this paper, we study the algebraic structure of $(σ,δ)$-polycyclic codes, defined as submodules in the quotient module $S/Sf$, where $S=R[x,σ,δ]$ 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 $(σ,δ)$-polycyclic codes are $(σ,δ)$-sequential codes. By using $(σ,δ)$-Pseudo Linear Transformation, we define the annihilator dual of $(σ,δ)$-polycyclic codes. Then, we demonstrate that the annihilator duals of $(σ,δ)$-polycyclic codes maintain their $(σ,δ)$-polycyclic nature. Furthermore, we classify when two $(σ,δ)$-polycyclic codes are Hamming isometrical equivalent. By employing Wedderburn polynomials, we introduce simple-root $(σ,δ)$-polycyclic codes. Subsequently, we define the $(σ, δ)$-Mattson-Solomon transform for this class of codes and we address the problem of decomposing these codes by using the properties of Wedderburn polynomials. |
| title | $(σ,δ)$-polycyclic codes in Ore extensions over rings |
| topic | Information Theory |
| url | https://arxiv.org/abs/2312.07193 |