Constructions of Quantum Codes Based on Quadratic Residues

In this paper, we illustrate how to simplify the constructions of quantum error-correction codes via the quadratic residues. The suggested quantum code, which is the stabilizer quantum code generated from an Abelian group with the basis being commutative quantum operators selected from rows of an Pauli block matrix, does not require the dual-containing (or self-orthogonal) constraint necessary for the standard quantum error-correction code, thus allowing us to construct a quantum code with the large codeword length.