On $(\theta ,\Theta )$-cyclic codes and their applications in constructing QECCs

Abstract

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 ${\mathbb{F}}_{q}R.$ Specifically, we analyze the structure of these codes as left $R[x:\Theta ]$-submodules of ${\mathfrak{R}}_{r,s}=\frac{{\mathbb{F}}_q[x:\theta ]}{⟨x^r-1⟩}\times \frac{R[x:\Theta ]}{⟨x^s-1⟩}$. 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 ${\mathbb{F}}_{q}R$ 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 ${\mathbb{F}}_{q}R$. We support our theoretical results with illustrative examples.

30 pages, 4 tables