The Structure of Constacyclic Codes of Length $2p^s$ over Finite Chain Ring

Wateekorn Sriwirach, Chakkrid Klin-eam

Abstract

Let $R$ be a finite commutative chain ring with identity of characteristic $p^a$ that has maximal ideal $\langle z\rangle$. In this paper, we study $\lambda$-constacyclic codes of length $2p^s$ over the ring $R$, for any unit $\lambda$ of $R$. If the unit $\lambda$ is not a square, the rings $\mathcal{R}_{\lambda}=\frac{R[x]}{\langle x^{2p^s}-\lambda\rangle}$ is a local ring with maximal ideal $\langle x^2-r, z\rangle$, where $r\in R$ such that $\lambda-r^{p^s}$ is not invertible. When there exists a unit  $\lambda_0$ of $R$ such that $\lambda=\lambda^{p^s}_0$, we prove that $x^2-\lambda_0$ is nilpotent with nilpotency index $ap^s-(a-1)p^{s-1}$. When $\lambda=\lambda^{p^s}_0+z\omega$, for some unit $\omega$ of $R$, we show that $\mathcal{R}_{\lambda}$ is also a chain ring with maximal ideals $\langle x^2-\lambda_0\rangle$. Moreover, when $\lambda=\lambda^{p^s}_0+z^2\omega$, it shown that $\mathcal{R}_{\lambda}$ is a chain ring with maximal ideals $\langle x^2-\lambda_0\rangle$. Furthermore, the algebraic structure and dual of all $\lambda$-constacyclic codes are obtained.