Repeated-root constacyclic codes of prime power lengths over finite chain rings

We study the algebraic structure of repeated-root λ  -constacyclic codes of prime power length psps over a finite commutative chain ring R   with maximal ideal 〈γ〉〈γ〉. It is shown that, for any unit λ of the chain ring R  , there always exists an element r∈Rr∈R such that λ−rpsλ−rps is not invertible, and furthermore, the ambient ring R[x]〈xps−λ〉 is a local ring with maximal ideal 〈x−r,γ〉〈x−r,γ〉. When there is a unit λ0λ0 such that λ=λ0ps, the nilpotency index of x−λ0x−λ0 in the ambient ring R[x]〈xps−λ〉 is established. When λ=λ0ps+γw, for some unit w of R  , it is shown that the ambient ring R[x]〈xps−λ〉 is a chain ring with maximal ideal 〈xps−λ0〉〈xps−λ0〉, which in turn provides structure and sizes of all λ-constacyclic codes and their duals. Among other things, situations when a linear code over R is both α- and β-constacyclic, for different units α, β, are discussed.