A Gröbner Approach to Dual-Containing Cyclic Left Module $(θ,δ)$-Codes over Finite Commutative Frobenius Rings

Abstract

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 $4$.