RT distance and weight distributions of Type 1 constacyclic codes of length\\ $4p^s$ over $\frac{\mathbb F_{p^m}[u]}{\left\langle u^a \right\rangle}$

Authors: HAI DINH, BAC NGUYEN, SONGSAK SRIBOONCHITTA

Abstract: For any odd prime $p$ such that $p^m \equiv 1 \pmod{4}$, the class of $\Lambda$-constacyclic codes of length $4p^s$ over the finite commutative chain ring ${\cal R}_a=\frac{\mathbb F_{p^m}[u]}{\left\langle u^a \right\rangle}=\mathbb F_{p^m} + u \mathbb F_{p^m}+ \dots + u^{a-1}\mathbb F_{p^m}$, 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 $\Lambda_0, \Lambda_1, \dots, \Lambda_{a-1} \in \mathbb F_{p^m}$, $\Lambda_0 \,{\not=}\, 0, \, \Lambda_1 \,{\not=}\, 0$, is investigated. If the unit $\Lambda$ is a square, each $\Lambda$-constacyclic code of length $4p^s$ is expressed as a direct sum of a $-\lambda$-constacyclic code and a $\lambda$-constacyclic code of length $2p^s$. In the main case that the unit $\Lambda$ is not a square, we show that any nonzero polynomial of degree $<4$ over $\mathbb F_{p^m}$ is invertible in the ambient ring $\frac{{\cal R}_a[x]}{\langle x^{4p^s}-\Lambda\rangle}$ and use it to prove that the ambient ring $\frac{\mathcal R_a[x]}{\langle x^{4p^s}-\Lambda\rangle}$ is a chain ring with maximal ideal $\langle x^4-\lambda_0 \rangle$, where $\lambda_0^{p^s}=\Lambda_0.$ As an application, the number of codewords and the dual of each $\lambda$-constacyclic code are provided. Furthermore, we get the Rosenbloom--Tsfasman (RT) distance and weight distributions of such codes. Using these results, the unique MDS code with respect to the RT distance is identified.

Keywords: RT distance, constacyclic codes, dual codes, chain rings

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