$\mathbb {F}_p$-Linear and $\mathbb {F}_{p^m}$-Linear Qudit Codes From Dual-Containing Classical Codes

Abstract

Quantum code construction from two classical codes $D_1[n,k_1,d_1]$ and $D_2[n,k_2,d_2]$ over the field $\mathbb {F}_{p^m}$ ( $p$ is prime and $m$ is an integer) satisfying the dual containing criteria $D_1^{\perp } \subset D_2$ 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 $\mathbb {F}_{p}$ -linear CSS codes and the well-known $\mathbb {F}_{p^m}$ -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 $\mathbb {F}_{p}$ -linear and $\mathbb {F}_{p^m}$ -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 $\mathbb {F}_{p^m}$ -linear CSS codes, we provide the explicit form of the check matrix and show that the minimum distances $d_x$ and $d_z$ are equal to $d_2$ and $d_1$ , respectively, if and only if the code is nondegenerate. 3) We propose two classes of quantum codes obtained from the codes $D_1$ and $D_2$ , where one code is an $\mathbb {F}_{p^l}$ -linear code ( $l$ divides $m$ ) and the other code is obtained from a particular subgroup of the stabilizer group of the $\mathbb {F}_{p^m}$ -linear CSS code. Within each class of codes, we demonstrate the tradeoff between higher rates and better error detection and correction capability.