A Gröbner Approach to Dual-Containing Cyclic Left Module (θ,δ)-Codes Rg/Rf⊂ R/Rf over Finite Commutative Frobenius Rings
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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 𝒞=Rg/Rf⊂ 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,…,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 𝒞=Rg/Rf⊂ 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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