关键词: Finite chain rings, formal power series rings, cyclic codes, tower of codes, Hensel lift

Abstract:

Let R be a finite chain ring with maximal ideal （γ） and residue field F, 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 : γ^t−1) can be associated with C, where for any r ∈ R, (C : r) = {e ∈ Rⁿ | re ∈ C}. Using generator elements of the projection of such a tower of codes to the residue field F, 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 F. Furthermore, we obtain a characterization of cyclic codes over the formal power series ring of a finite chain ring.

Key words: Finite chain rings, formal power series rings, cyclic codes, tower of codes, Hensel lift