Tallennettuna:
| Päätekijät: | , , , , |
|---|---|
| Aineistotyyppi: | Preprint |
| Julkaistu: |
2024
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| Aiheet: | |
| Linkit: | https://arxiv.org/abs/2404.00613 |
| Tagit: |
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Sisällysluettelo:
- Let $\mathbb F_q$ be a finite field, where $q$ is an odd prime power. Let $R=\mathbb{F}_q+u\mathbb{F}_q+v\mathbb{F}_q+uv\mathbb F_q$ with $u^2=u,v^2=v,uv=vu$. In this paper, we study the algebraic structure of $(θ, Θ)$-cyclic codes of block length $(r,s )$ over $\mathbb{F}_qR.$ Specifically, we analyze the structure of these codes as left $R[x:Θ]$-submodules of $\mathfrak{R}_{r,s} = \frac{\mathbb{F}_q[x:θ]}{\langle x^r-1\rangle} \times \frac{R[x:Θ]}{\langle x^s-1\rangle}$. Our investigation involves determining generator polynomials and minimal generating sets for this family of codes. Further, we discuss the algebraic structure of separable codes. A relationship between the generator polynomials of $(θ, Θ)$-cyclic codes over $\mathbb F_qR$ and their duals is established. Moreover, we calculate the generator polynomials of dual of $(θ, Θ)$-cyclic codes. As an application of our study, we provide a construction of quantum error-correcting codes (QECCs) from $(θ, Θ)$-cyclic codes of block length $(r,s)$ over $\mathbb{F}_qR$. We support our theoretical results with illustrative examples.