Contraction of cyclic codes over finite chain rings

In this paper, R is a finite chain ring with residue field Fq and γ is a unit in R. By assuming that the multiplicative order u of γ is coprime to q, we give the trace-representation of any simple-root γ-constacyclic code over R of length ℓ, and on the other hand show that any cyclic code over R of length uℓ is a direct sum of trace-representable cyclic codes. Finally, we characterize the simple-root, contractable and cyclic codes over R of length uℓ into γ-constacyclic codes of length ℓ.