This thesis is about the quantum error-correcting codes. Although there are many methods to construct them, finding a good quantum code is still a complicated and difficult task, for we do not know which methods to use. To solve this problem, we did a thorough research on a special class of quantum codes--additive cyclic quantum codes. A new search algorithm has been designed and a lot of good additive cyclic quantum codes found by this algorithm are presented in this thesis. By showing the success of this algorithm and great value of additive cyclic quantum codes, we have greatly reduced the complexity of finding good quantum codes. Quantum error-correction theory is key part of quantum information theory. As quantum information theory is a quite new field which has been developing fast during the last decade, we spent much time explaining the primary concepts of it. We introduced linear algebra, quantum mechanics, quantum operations, quantum error-correction theory and quantum error-correcting codes. In a word, this thesis serves two functions: (1) an exploration of quantum error-correcting codes and (2) an introduction to quantum information theory
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