Bull. Korean Math. Soc. 2018; 55(4): 1189-1208
Online first article April 3, 2018 Printed July 31, 2018
https://doi.org/10.4134/BKMS.b170659
Copyright © The Korean Mathematical Society.
Hai Q. Dinh, Bac Trong Nguyen, Songsak Sriboonchitta
Ton Duc Thang University, Thai Nguyen University, Chiang Mai University
The aim of this paper is to study the class of $\Lambda$-constacyclic codes of length $2p^s$ over the finite commutative chain ring ${\mathcal R}_a=\frac{\mathbb F_{p^m}[u]}{\left\langle u^a \right\rangle}=\mathbb F_{p^m} + u \mathbb F_{p^m}+ \dots + u^{a-1}\mathbb F_{p^m}$, for all units $\Lambda$ of $\mathcal R_a$ that have the form $\Lambda=\Lambda_0+u\Lambda_1+\dots+u^{a-1}\Lambda_{a-1}$, where $\Lambda_0, \Lambda_1, \dots, \Lambda_{a-1} \in \mathbb F_{p^m}$, $\Lambda_0 \,{\not=}\, 0, \, \Lambda_1 \,{\not=}\, 0$. The algebraic structure of all $\Lambda$-constacyclic codes of length $2p^s$ over ${\mathcal R}_a$ and their duals are established. As an application, this structure is used to determine the Rosenbloom-Tsfasman (RT) distance and weight distributions of all such codes. Among such constacyclic codes, the unique MDS code with respect to the RT distance is obtained.
Keywords: constacyclic codes, dual codes, chain rings
MSC numbers: Primary 94B15, 94B05; Secondary 11T71
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