On $(\theta, \Theta)$-cyclic codes and their applications in constructing QECCs
<|reference_start|>On $(\theta, \Theta)$-cyclic codes and their applications in constructing QECCs: 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 $(\theta, \Theta)$-cyclic codes of block length $(r,s )$ over $\mathbb{F}_qR.$ Specifically, we analyze the structure of these codes as left $R[x:\Theta]$-submodules of $\mathfrak{R}_{r,s} = \frac{\mathbb{F}_q[x:\theta]}{\langle x^r-1\rangle} \times \frac{R[x:\Theta]}{\langle x^s-1\rangle}$. 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 $(\theta, \Theta)$-cyclic codes over $\mathbb F_qR$ and their duals is established. Moreover, we calculate the generator polynomials of dual of $(\theta, \Theta)$-cyclic codes. As an application of our study, we provide a construction of quantum error-correcting codes (QECCs) from $(\theta, \Theta)$-cyclic codes of block length $(r,s)$ over $\mathbb{F}_qR$. We support our theoretical results with illustrative examples.<|reference_end|>
@article{shukla2024on,
    title={On $(\theta, \Theta)$-cyclic codes and their applications in
  constructing QECCs},
    author={Awadhesh Kumar Shukla, Sachin Pathak, Om Prakash Pandey, Vipul Mishra
  and Ashish Kumar Upadhyay},
    journal={arXiv preprint arXiv:2404.00613},
    year={2024},
    archivePrefix={arXiv},
    eprint={2404.00613},
    primaryClass={cs.IT math.IT}
}