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# Cyclic codes over rings of matrices

• *Corresponding author: Manoj Kumar Singh

Author's names are in alphabetical order.
This paper is a part of PhD thesis of Manoj Kumar Singh.

• In this paper, we consider the ring of matrices $\mathcal{A}$ of order $2$ over the ring $\mathbb{F}_2 [u] / \langle u^k \rangle$, where $u$ is an indeterminate with $u^k = 0$, i.e. $\mathcal{A} = M_2 ( \mathbb{F}_2 [u] / \langle u^k \rangle)$. We derive the structure theorem for cyclic codes of odd length $n$ over the ring $\mathcal{A}$ with the help of isometry map from $\mathcal{A}$ to $\mathbb{F}_4 [u, v] / \langle u^k, v^2, u v - v u \rangle$, where $v$ is an indeterminate satisfying $v^2 = 0$ and $u v = v u$. We define a map $\theta$ which takes the linear codes of odd length $n$ over $\mathcal{A}$ to linear codes of even length $2 k n$ over $\mathbb{F}_4$. We also define a weight on the ring $\mathcal{A}$ which is an extension of the weight defined over the ring $M_2 ( \mathbb{F}_2)$. An example is also given as applications to construct the linear codes of odd length $n$ over $\mathcal{A}$.

Mathematics Subject Classification: Primary: 94B05, 94B15; Secondary: 94B60.

 Citation:

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