A Gröbner Approach to Dual-Containing Cyclic Left Module $(θ,δ)$-Codes over Finite Commutative Frobenius Rings
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arXiv
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| Auteurs principaux: | , , |
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| Format: | Preprint |
| Publié: |
2023
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| _version_ | 1866916312613126144 |
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| author | Liu, Hedongliang Ott, Cornelia Ulmer, Felix |
| author_facet | Liu, Hedongliang Ott, Cornelia Ulmer, Felix |
| contents | For a skew polynomial ring $R=A[X;θ,δ]$ where $A$ is a commutative Frobenius ring, $θ$ an endomorphism of $A$ and $δ$ a $θ$-derivation of $A$, we consider cyclic left module codes $\mathcal{C}=Rg/Rf\subset R/Rf$ where $g$ is a left and right divisor of $f$ in $R$. In this paper, we derive a parity check matrix when $A$ is a finite commutative Frobenius ring using only the framework of skew polynomial rings. We consider rings $A=B[a_1,\ldots,a_s]$ which are free $B$-modules where the restriction of $δ$ and $θ$ to $B$ are polynomial maps. If a Gröbner basis can be computed over $B$, then we show that all Euclidean and Hermitian dual-containing codes $\mathcal{C}=Rg/Rf\subset R/Rf$ can be computed using a Gröbner basis. We also give an algorithm to test if the dual code is again a cyclic left module code. We illustrate our approach for rings of order $4$ with non-trivial endomorphism and the Galois ring of characteristic $4$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2308_13395 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | A Gröbner Approach to Dual-Containing Cyclic Left Module $(θ,δ)$-Codes over Finite Commutative Frobenius Rings Liu, Hedongliang Ott, Cornelia Ulmer, Felix Information Theory Discrete Mathematics Quantum Algebra Rings and Algebras For a skew polynomial ring $R=A[X;θ,δ]$ where $A$ is a commutative Frobenius ring, $θ$ an endomorphism of $A$ and $δ$ a $θ$-derivation of $A$, we consider cyclic left module codes $\mathcal{C}=Rg/Rf\subset R/Rf$ where $g$ is a left and right divisor of $f$ in $R$. In this paper, we derive a parity check matrix when $A$ is a finite commutative Frobenius ring using only the framework of skew polynomial rings. We consider rings $A=B[a_1,\ldots,a_s]$ which are free $B$-modules where the restriction of $δ$ and $θ$ to $B$ are polynomial maps. If a Gröbner basis can be computed over $B$, then we show that all Euclidean and Hermitian dual-containing codes $\mathcal{C}=Rg/Rf\subset R/Rf$ can be computed using a Gröbner basis. We also give an algorithm to test if the dual code is again a cyclic left module code. We illustrate our approach for rings of order $4$ with non-trivial endomorphism and the Galois ring of characteristic $4$. |
| title | A Gröbner Approach to Dual-Containing Cyclic Left Module $(θ,δ)$-Codes over Finite Commutative Frobenius Rings |
| topic | Information Theory Discrete Mathematics Quantum Algebra Rings and Algebras |
| url | https://arxiv.org/abs/2308.13395 |