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.

Full Text: PDF

Refbacks

  • There are currently no refbacks.


The Thai Journal of Mathematics organized and supported by The Mathematical Association of Thailand and Thailand Research Council and the Center for Promotion of Mathematical Research of Thailand (CEPMART).

Copyright 2020 by the Mathematical Association of Thailand.

All rights reserve. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, without the prior permission of the Mathematical Association of Thailand.

|ISSN 1686-0209|