$\mathbb {F}_p$-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 $m$ an integer) satisfying dual containing criteria $D_1^{\perp } \subset D_2$ using Calderbank–Shor–Steane (CSS) framework well-studied. We show that generalization of CSS for qubits to qudits yields different classes codes, namely, {F}_{p}$ -linear well-known based on check matrix-based definition coset-based qubits. Our contribution this article are three-folds. 1) study properties demonstrate tradeoff designing with higher rates or better error detection correction capability, useful quantum systems. 2) For we provide explicit form matrix minimum distances $d_x$ $d_z$ equal $d_2$ $d_1$ , respectively, if only nondegenerate. 3) propose obtained $D_1$ $D_2$ where one {F}_{p^l}$ $l$ divides ) other a particular subgroup stabilizer group code. Within each class between capability.