A study on constacyclic codes over the ring $\mathbb{Z}_4+u\mathbb{Z}_4+u^2\mathbb{Z}_4$

Document Type : Research Paper

Authors

1 Department of Mathematics, Manipur University, Imphal, Manipur-795003, India.

2 Department of Mathematics, D. M. College of Science, Imphal, Manipur-795001, India.

Abstract

This paper studies $\lambda$-constacyclic codes and skew $\lambda$-constacyclic codes over the finite commutative non-chain ring $R=\mathbb{Z}_4+u\mathbb{Z}_4+u^2\mathbb{Z}_4$ with $u^3=0$ for $\lambda= (1+2u+2u^2)$ and $(3+2u+2u^2)$. We introduce distinct Gray maps and show that the Gray images of $\lambda$-constacyclic codes are cyclic, quasi-cyclic, and permutation equivalent to quasi-cyclic codes over $\mathbb{Z}_4$. It is also shown that the Gray images of skew $\lambda$-constacyclic codes are quasi-cyclic codes of length $2n$ and index 2 over $\mathbb{Z}_4$. Moreover, the structure of $\lambda$-constacyclic codes of odd length $n$ over the ring $R$ is determined and give some suitable examples.

Keywords

References

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Algebraic Structures and Their Applications
Volume 11, Issue 2
May 2024
Pages 131-150

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