A Gröbner approach to dual-containing cyclic left module <inline-formula><tex-math id="M1">$ (\theta,\delta) $</tex-math></inline-formula>-codes over finite commutative Frobenius rings

For a skew polynomial ring $ R = A[X;\theta, \delta] $ where $ A $ is a commutative Frobenius ring, $ \theta $ an endomorphism of $ A $ and $ \delta $ a $ \theta $-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 $ \delta $ and $ \theta $ 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 $.