Quantum computers theoretically are able to solve certain problems more quickly than any deterministic or probabilistic computers. A quantum computer exploits the rules of quantum mechanics to speed up computations. However, one has to mitigate the resulting noise and decoherence effects to avoid computational errors in order to successfully build quantum computers. In this paper, we construct asymmetric quantum codes to protect quantum information over asymmetric quantum channels, Pr Z ≥ Pr X. Two generic methods are presented to derive asymmetric quantum cyclic codes using the generator polynomials and defining sets of classical cyclic codes. Consequently, the methods allow us to construct several families of quantum BCH, RS, and RM codes over asymmetric quantum channels. Finally, the methods are used to construct families of asymmetric subsystem codes.