Repeated-root constacyclic codes of length 2p^s over $\mathbb {F}_{p^{m}}+u\mathbb {F}_{p^{m}}+u^{2}\mathbb {F}_{p^{m}}$

Abstract

Let λ be a unit of the finite commutative chain ring $R=\mathbb {F}_{p^{m}}+u\mathbb {F}_{p^{m}}+u^{2}\mathbb {F}_{p^{m}}=\{\alpha +u\upbeta +u^{2}\gamma : \alpha ,\upbeta ,\gamma \in \mathbb { F}_{p^{m}}\}$ with u³ = 0, where p is an odd prime and m is a positive integer. In this paper, we consider any λ-constacyclic codes of length 2p^s over R. In the case of square λ = δ², where δ ∈ R, the algebraic structures of all λ-constacyclic codes of length 2p^s over R are determined by the Chinese Remainder Theorem in terms of polynomial generators. Precisely, each λ-constacyclic code of length 2p^s is represented as a direct sum of a (−δ)-constacyclic code and a δ-constacyclic code of length p^s. In the case of non-square λ = α + uβ + u²γ or λ = α + uβ, where $\alpha ,\upbeta ,\gamma \in \mathbb {F}_{p^{m}}\setminus \{0\}$, it is shown that the ring $\mathcal {R}=\frac {R[x]}{\langle x^{2p^{s}}-\lambda \rangle }$ is a chain ring. In the case of non-square $\lambda =\alpha \in \mathbb {F}_{p^{m}}\setminus \{0\}$, it turns out that λ-constacyclic codes are classified into 8 distinct types of ideals, and the detailed structures of ideals in each type are provided.