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

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.