A Gröbner Approach to Dual-Containing Cyclic Left Module $(\hat{I}¸,\hat{I}´)$-Codes over Finite Commutative Frobenius Rings

Abstract

For a skew polynomial ring $R=A[X;\hat{I}¸,\hat{I}´]$ where $A$ is a commutative Frobenius ring, $\hat{I}¸$ an endomorphism of $A$ and $\hat{I}´$ a $\hat{I}¸$-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 $A=B[a_1,\dots ,a_s]$ which are free $B$-modules where the restriction of $\hat{I}´$ and $\hat{I}¸$ 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 $4$.

20 pages, 11 tables