A Gröbner Approach to Dual-Containing Cyclic Left Module $(θ,δ)$-Codes $Rg/Rf\subset R/Rf$ over Finite Commutative Frobenius Rings - Archive ouverte HAL Accéder directement au contenu
Pré-Publication, Document De Travail Année : 2023

A Gröbner Approach to Dual-Containing Cyclic Left Module $(θ,δ)$-Codes $Rg/Rf\subset R/Rf$ over Finite Commutative Frobenius Rings

Résumé

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 $B$-algebras 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 $4$.
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Dates et versions

hal-04382332 , version 1 (09-01-2024)

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Cornelia Ott, Félix Ulmer, Hedongliang Liu. A Gröbner Approach to Dual-Containing Cyclic Left Module $(θ,δ)$-Codes $Rg/Rf\subset R/Rf$ over Finite Commutative Frobenius Rings. 2024. ⟨hal-04382332⟩
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