Quantum error-correcting codes (QECCs) will be the ultimate enabler of future quantum computing and quantum information processing. Stabilizer codes are the most important class of QECCs since the first discovery of QECCs in the mid-1990s. In this thesis, we study the design of QECCs and provide several contributions to quantum stabilizer code constructions. The first contribution is the design of families of quantum stabilizer codes using quadratic residues (QR) sets and difference sets. We study the distance property and dimension for the families of quantum stabilizer codes constructed from QR sets. We give three design criteria for constructing quantum stabilizer codes from difference sets. We show that using the subsets of difference sets can further improve the proposed code performance. We then design families of quantum low-density parity-check (LDPC) codes from classical quasi-cyclic LDPC codes for large-scale quantum systems. The proposed quantum LDPC codes of quasi-cyclic structure and various code rates are constructed from a family of proto-graph LDPC codes based on the QR set and Latin square. We provide two constructions based on the adjunction and concatenation of a proto-matrix and one construction based on the unique transformation of a proto-matrix. We derive the dimension of the proposed quantum LDPC codes and provide a lower bound for its minimum distance. The performance of the proposed quantum LDPC codes over quantum depolarizing channels with iterative sum-product decoding algorithms is illustrated. Furthermore, we propose a construction of quantum LDPC codes with rate at least 0.9 by performing tensor product operation between two non-binary parity-check matrices obtained from the idempotent polynomials of QR/NQR sets. Next, we study quantum synchronizable codes that correct both quantum noise and block synchronization errors. We propose a general construction of quantum synchronizable codes with CSS structure from classical chain-containing cyclic codes, and derive a distance bound using rational function for the proposed quantum synchronizable codes.We design a class of quantum synchronizable codes from classical quadratic residue codes over binary field. We show that these codes are a subclass of the proposed chain-containing cyclic codes, and their code length and dimension are equal to Mersenne prime and one, respectively. Lastly, inspired by the phenomenon of channel mismatch effect for classical LDPC codes, we investigate the effect of channel mismatch for quantum LDPC codes over quantum depolarizing channels. We show that the degraded performance due to the channel mismatch can be mitigated by introducing a weighted channel information into the iterative sum-product decoder.