Repeated-root constacyclic codes of prime power lengths over finite chain rings - Hai Q. Dinh (2018)
© 2016 Elsevier Inc. We study the algebraic structure of repeated-root λ-constacyclic codes of prime power length p s over a finite commutative chain ring R with maximal ideal 〈γ〉. It is shown that, for any unit λ of the chain ring R, there always exists an element r∈R such that λ−r p s is not invertible, and furthermore, the ambient ring R[x]〈xps−λ〉 is a local ring with maximal ideal 〈x−r,γ〉. When there is a unit λ 0 such that λ=λ 0 p s , the nilpotency index of x−λ 0 in the ambient ring R[x]〈xps−λ〉 is established. When λ=λ 0 p s +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 〈x p s −λ 0 〉, 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.