Asymmetric Quantum Cyclic Codes

It is recently conjectured in quantum information processing that phase-shift errors occur with high probability than qubit-flip errors, hence the former is more disturbing to quantum information than the later one. This leads us to construct asymmetric quantum error controlling codes to protect quantum information over asymmetric channels, $\Pr Z \geq \Pr X$. In this paper we present two generic methods 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 asymmetric quantum BCH, RS, and RM codes. Finally, the methods are used to construct families of asymmetric subsystem codes.