| Summary: | Quantum code construction from two classical codes <inline-formula><tex-math notation="LaTeX">$D_1[n,k_1,d_1]$</tex-math></inline-formula> and <inline-formula><tex-math notation="LaTeX">$D_2[n,k_2,d_2]$</tex-math></inline-formula> over the field <inline-formula><tex-math notation="LaTeX">$\mathbb {F}_{p^m}$</tex-math></inline-formula> (<inline-formula><tex-math notation="LaTeX">$p$</tex-math></inline-formula> is prime and <inline-formula><tex-math notation="LaTeX">$m$</tex-math></inline-formula> is an integer) satisfying the dual containing criteria <inline-formula><tex-math notation="LaTeX">$D_1^{\perp } \subset D_2$</tex-math></inline-formula> using the Calderbank–Shor–Steane (CSS) framework is well-studied. We show that the generalization of the CSS framework for qubits to qudits yields two different classes of codes, namely, the <inline-formula><tex-math notation="LaTeX">$\mathbb {F}_{p}$</tex-math></inline-formula>-linear CSS codes and the well-known <inline-formula><tex-math notation="LaTeX">$\mathbb {F}_{p^m}$</tex-math></inline-formula>-linear CSS codes based on the check matrix-based definition and the coset-based definition of CSS codes over qubits. Our contribution to this article are three-folds. 1) We study the properties of the <inline-formula><tex-math notation="LaTeX">$\mathbb {F}_{p}$</tex-math></inline-formula>-linear and <inline-formula><tex-math notation="LaTeX">$\mathbb {F}_{p^m}$</tex-math></inline-formula>-linear CSS codes and demonstrate the tradeoff for designing codes with higher rates or better error detection and correction capability, useful for quantum systems. 2) For <inline-formula><tex-math notation="LaTeX">$\mathbb {F}_{p^m}$</tex-math></inline-formula>-linear CSS codes, we provide the explicit form of the check matrix and show that the minimum distances <inline-formula><tex-math notation="LaTeX">$d_x$</tex-math></inline-formula> and <inline-formula><tex-math notation="LaTeX">$d_z$</tex-math></inline-formula> are equal to <inline-formula><tex-math notation="LaTeX">$d_2$</tex-math></inline-formula> and <inline-formula><tex-math notation="LaTeX">$d_1$</tex-math></inline-formula>, respectively, if and only if the code is nondegenerate. 3) We propose two classes of quantum codes obtained from the codes <inline-formula><tex-math notation="LaTeX">$D_1$</tex-math></inline-formula> and <inline-formula><tex-math notation="LaTeX">$D_2$</tex-math></inline-formula>, where one code is an <inline-formula><tex-math notation="LaTeX">$\mathbb {F}_{p^l}$</tex-math></inline-formula>-linear code (<inline-formula><tex-math notation="LaTeX">$l$</tex-math></inline-formula> divides <inline-formula><tex-math notation="LaTeX">$m$</tex-math></inline-formula>) and the other code is obtained from a particular subgroup of the stabilizer group of the <inline-formula><tex-math notation="LaTeX">$\mathbb {F}_{p^m}$</tex-math></inline-formula>-linear CSS code. Within each class of codes, we demonstrate the tradeoff between higher rates and better error detection and correction capability.
|
|---|
