Quantum code construction from two classical codes D1[n,k1,d1] and D2[n,k2,d2] over the field F_p^m ( p is prime and m is an integer) satisfying the dual containing criteria D⊥1⊂D2 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 F_p -linear CSS codes and the well-known Fpm -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 F_p -linear and 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 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 d₂ and d₁ , respectively, if and only if the code is nondegenerate. 3) We propose two classes of quantum codes obtained from the codes D₁ and D₂ , where one code is an F_p^l -linear code ( l divides m ) and the other code is obtained from a particular subgroup of the stabilizer group of the 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.

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https://ieeexplore.ieee.org/document/9431703