A Study of Cyclic and Constacyclic Codes over $\mathbb{Z}_{4}+u_{2}\mathbb{Z}_{4}+u_{3}\mathbb{Z}_{4}+\ldots+u_{t}\mathbb{Z}_{4}$

Articles in Press

Document Type : Original paper

Authors

1 School of Mathematical Sciences, SRTM University, Nanded, India

2 Department of Mathematics, Shri Guru Gobind Singhji Institute of Engineering and Technology, Nanded, India

3 Shri Sant Gadge Maharaj Mahavidyalaya, Loha, Nanded, India

Abstract

Constacyclic codes constitute a significant class of linear codes in coding theory and play a crucial role in the construction of optimal codes. Several optimal linear codes have been derived from constacyclic codes. In 2015, Ashraf and Mohammad investigated $(1+2u)$-constacyclic codes over $\mathbb{Z}_{4}+u\mathbb{Z}_{4}$ with $u^{2}=0$. More recently, G. Karthick studied $(1+2u+2v)$-constacyclic codes over the semi-local ring $\mathbb{Z}_{4}+u\mathbb{Z}_{4}+v\mathbb{Z}_{4}$ under the conditions $u^{2}=3u$, $v^{2}=3v$, and $uv=vu=0$. In this paper, we generalize their results by examining $(1+2u_{2}+2u_{3}+\dots+2u_{t})$-constacyclic codes over the semi-local ring $\mathcal{S} = \mathbb{Z}_{4} + u_{2} \mathbb{Z}_{4} + u_{3} \mathbb{Z}_{4} + \dots + u_{t} \mathbb{Z}_{4}$, where $u_{i}^{2} = k u_{i}$ and $u_{i} u_{j} = u_{j} u_{i} = 0$ for $2 \leq i \leq t$, $i \neq j$, with $u_{1}=1$ and $k \in \mathbb{Z}_{4}$. We focus on $(1+2u_{2}+2u_{3}+\dots+2u_{t})$-constacyclic codes over $\mathcal{S}$ and establish their structural properties. By introducing new Gray maps, we demonstrate that these constacyclic codes can be transformed into cyclic and quasi-cyclic codes over $\mathbb{Z}_{4}$. Furthermore, we characterize a generating set for these codes when the code length is odd. %nand provide explicit examples illustrating their construction. Our findings contribute to the database of $mathbb{Z}_{4}$ codes and enhance the understanding of constacyclic codes over extended non-chain rings.

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References

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Communications in Combinatorics and Optimization

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Available Online from 20 November 2025

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