Skew Generalized Polycyclic Codes with Derivations
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arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2019
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| _version_ | 1866913635189653504 |
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| author | Patel, Shikha Prakash, Om |
| author_facet | Patel, Shikha Prakash, Om |
| contents | In this paper, we first consider the iterated skew polynomial ring $\mathscr{R}[z_1;τ_1,δ_{τ_1}]$\\$[z_2;τ_2,δ_{τ_2}]$, where $\mathscr{R}$ is a finite ring with unity. Then we use this structure for the construction of skew generalized polycyclic codes over the ring $\mathscr{R}$ and finite field $\mathbb{F}_q$, where $q=p^m$ for some positive integer $m$. Further, we derive the structure of the generator and parity check matrices for skew generalized polycyclic codes. Furthermore, we improve the Bose-Chaudhuri-Hocquenghem (BCH) lower bound for a minimum distance of skew generalized polycyclic codes with non-zero derivations over a finite field. Moreover, we find a sufficient condition for a code to be a maximum-distance-separable (MDS) code. In addition, we provide examples of MDS codes to show the importance of our results. A comparative summary of our work with other linear codes is also discussed. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1907_06086 |
| institution | arXiv |
| publishDate | 2019 |
| record_format | arxiv |
| spellingShingle | Skew Generalized Polycyclic Codes with Derivations Patel, Shikha Prakash, Om Information Theory 12Y05, 16Z05, 94B05, 94B35 In this paper, we first consider the iterated skew polynomial ring $\mathscr{R}[z_1;τ_1,δ_{τ_1}]$\\$[z_2;τ_2,δ_{τ_2}]$, where $\mathscr{R}$ is a finite ring with unity. Then we use this structure for the construction of skew generalized polycyclic codes over the ring $\mathscr{R}$ and finite field $\mathbb{F}_q$, where $q=p^m$ for some positive integer $m$. Further, we derive the structure of the generator and parity check matrices for skew generalized polycyclic codes. Furthermore, we improve the Bose-Chaudhuri-Hocquenghem (BCH) lower bound for a minimum distance of skew generalized polycyclic codes with non-zero derivations over a finite field. Moreover, we find a sufficient condition for a code to be a maximum-distance-separable (MDS) code. In addition, we provide examples of MDS codes to show the importance of our results. A comparative summary of our work with other linear codes is also discussed. |
| title | Skew Generalized Polycyclic Codes with Derivations |
| topic | Information Theory 12Y05, 16Z05, 94B05, 94B35 |
| url | https://arxiv.org/abs/1907.06086 |