On a class of constacyclic codes of length $2p^s$ over $\frac{\mathbb F_{p^m}[u]}{\left\langle u^a \right\rangle}$

Bull. Korean Math. Soc. 2018 Vol. 55, No. 4, 1189-1208 https://doi.org/10.4134/BKMS.b170659 Published online April 3, 2018 Printed July 31, 2018

Hai Q. Dinh, Bac Trong Nguyen, Songsak Sriboonchitta Ton Duc Thang University, Thai Nguyen University, Chiang Mai University

Abstract : 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.