Skew cyclic codes over $\mathbb{Z}_4+v\mathbb{Z}_4$ with derivation: structural properties and computational results
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| Format: | Preprint |
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2021
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| _version_ | 1866911763271778304 |
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| author | Suprijanto, Djoko Tang, Hopein Christofen |
| author_facet | Suprijanto, Djoko Tang, Hopein Christofen |
| contents | In this work, we study a class of skew cyclic codes over the ring $R:=\mathbb{Z}_4+v\mathbb{Z}_4,$ where $v^2=v,$ with an automorphism $θ$ and a derivation $Δ_θ,$ namely codes as modules over a skew polynomial ring $R[x;θ,Δ_θ],$ whose multiplication is defined using an automorphism $θ$ and a derivation $Δ_θ.$ We investigate the structures of a skew polynomial ring $R[x;θ,Δ_θ].$ We define $Δ_θ$-cyclic codes as a generalization of the notion of cyclic codes. The properties of $Δ_θ$-cyclic codes as well as dual $Δ_θ$-cyclic codes are derived. As an application, some new linear codes over $\mathbb{Z}_4$ with good parameters are obtained by Plotkin sum construction, also via a Gray map as well as residue and torsion codes of these codes. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2110_01580 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Skew cyclic codes over $\mathbb{Z}_4+v\mathbb{Z}_4$ with derivation: structural properties and computational results Suprijanto, Djoko Tang, Hopein Christofen Information Theory 94B05, 94B15, 11T71 In this work, we study a class of skew cyclic codes over the ring $R:=\mathbb{Z}_4+v\mathbb{Z}_4,$ where $v^2=v,$ with an automorphism $θ$ and a derivation $Δ_θ,$ namely codes as modules over a skew polynomial ring $R[x;θ,Δ_θ],$ whose multiplication is defined using an automorphism $θ$ and a derivation $Δ_θ.$ We investigate the structures of a skew polynomial ring $R[x;θ,Δ_θ].$ We define $Δ_θ$-cyclic codes as a generalization of the notion of cyclic codes. The properties of $Δ_θ$-cyclic codes as well as dual $Δ_θ$-cyclic codes are derived. As an application, some new linear codes over $\mathbb{Z}_4$ with good parameters are obtained by Plotkin sum construction, also via a Gray map as well as residue and torsion codes of these codes. |
| title | Skew cyclic codes over $\mathbb{Z}_4+v\mathbb{Z}_4$ with derivation: structural properties and computational results |
| topic | Information Theory 94B05, 94B15, 11T71 |
| url | https://arxiv.org/abs/2110.01580 |