On some classes of constacyclic codes over polynomial residue rings

Hai Q. Dinh; Hien D. T. Nguyen

doi:10.3934/amc.2012.6.175

Article Contents

2012, Volume 6, Issue 2: 175-191. Doi: 10.3934/amc.2012.6.175

This issue Previous Article Combinatorial batch codes: A lower bound and optimal constructions Next Article Double-circulant and bordered-double-circulant constructions for self-dual codes over $R_2$

On some classes of constacyclic codes over polynomial residue rings

Hai Q. Dinh^1, and
Hien D. T. Nguyen^1,

1.
Department of Mathematical Sciences, Kent State University, 4314 Mahoning Avenue, Warren, Ohio 44483, USA, and Department of Mathematics, Vinh University, Vinh

Received: March 31, 2011

Revised: June 30, 2011

Published: April 2012

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Abstract

The polynomial residue ring $\mathcal R_a=\frac{\mathbb F_{2^m}[u]}{\langle u^a \rangle}=\mathbb F_{2^m} + u \mathbb F_{2^m}+ \dots + u^{a - 1}\mathbb F_{2^m}$ is a chain ring with residue field $\mathbb F_{2^m}$, that contains precisely $(2^m-1)2^{m(a-1)}$ units, namely, $\alpha_0+u\alpha_1+\dots+u^{a-1}\alpha_{a-1}$, where $\alpha_0,\alpha_1,\dots,\alpha_{a-1} \in \mathbb F_{2^m}$, $\alpha_0 \neq 0$. Two classes of units of $\mathcal R_a$ are considered, namely, $\lambda=1+u\lambda_1+\dots+u^{a-1}\lambda_{a-1}$, where $\lambda_1, \dots, \lambda_{a-1} \in \mathbb F_{2^m}$, $\lambda_1 \neq 0$; and $\Lambda=\Lambda_0+u\Lambda_1+\dots+u^{a-1}\Lambda_{a-1}$, where $\Lambda_0, \Lambda_1, \dots, \Lambda_{a-1} \in \mathbb F_{2^m}$, $\Lambda_0 \neq 0, \Lambda_1 \neq 0$. Among other results, the structure, Hamming and homogeneous distances of $\Lambda$-constacyclic codes of length $2^s$ over $\mathcal R_a$, and the structure of $\lambda$-constacyclic codes of any length over $\mathcal R_a$ are established.

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Mathematics Subject Classification: Primary: 94B15; Secondary: 12Y05.

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