On the characterization of cyclic codes over two classes of rings

Let R be a finite chain ring with maximal ideal 〈 γ 〉 and residue field , and let γ be of nilpotency index t. To every code C of length n over R, a tower of codes C = (C: γ0) ⊆ (C: γ) ⊆ … ⊆ (C:γi) ⊆ … ⊆ (C: γt−1) can be associated with C, where for any r ɛ R, (C:r) = {e ɛ Rn | re ɛ C}. Using generator elements of the projection of such a tower of codes to the residue field , we characterize cyclic codes over R. This characterization turns the condition for codes over R to be cyclic into one for codes over the residue field . Furthermore, we obtain a characterization of cyclic codes over the formal power series ring of a finite chain ring.