Skew Laurent Series and General Cyclic Convolutional Codes
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866917193413820416 |
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| author | Gómez-Torrecillas, José Sánchez-Hernández, José Patricio |
| author_facet | Gómez-Torrecillas, José Sánchez-Hernández, José Patricio |
| contents | Convolutional codes were originally conceived as vector subspaces of a finite-dimensional vector space over a field of Laurent series having a polynomial basis. Piret and Roos modeled cyclic structures on them by adding a module structure over a finite-dimensional algebra skewed by an algebra automorphism. These cyclic convolutional codes turn out to be equivalent to some right ideals of a skew polynomial ring built from the automorphism. When a skew derivation is considered, serious difficulties arise in defining such a skewed module structure on Laurent series. We discuss some solutions to this problem which involve a purely algebraic treatment of the left skew Laurent series built from a left skew derivation of a general coefficient ring, when possible. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_05022 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Skew Laurent Series and General Cyclic Convolutional Codes Gómez-Torrecillas, José Sánchez-Hernández, José Patricio Rings and Algebras 16U20, 16S36, 16W60, 94B10 Convolutional codes were originally conceived as vector subspaces of a finite-dimensional vector space over a field of Laurent series having a polynomial basis. Piret and Roos modeled cyclic structures on them by adding a module structure over a finite-dimensional algebra skewed by an algebra automorphism. These cyclic convolutional codes turn out to be equivalent to some right ideals of a skew polynomial ring built from the automorphism. When a skew derivation is considered, serious difficulties arise in defining such a skewed module structure on Laurent series. We discuss some solutions to this problem which involve a purely algebraic treatment of the left skew Laurent series built from a left skew derivation of a general coefficient ring, when possible. |
| title | Skew Laurent Series and General Cyclic Convolutional Codes |
| topic | Rings and Algebras 16U20, 16S36, 16W60, 94B10 |
| url | https://arxiv.org/abs/2507.05022 |