Constructions Of Quantum Error Correcting Codes Based On Cyclotomic Cosets

In quantum computation and quantum communication, there are the inevitable interaction between quantum bits and the external environment, leading to qubit decoherence. Quantum error correcting codes provide the most efficient way to overcome decoherence. In this dissertation, we take the cyclotomic cosets as the foundation, and research the construction of quantum error correcting codes. The contributions of the dissertation are outlined as follows:1. We construct symmetric quantum codes via classical BCH code and constacyclic code. We analyze the related properties of the cyclotomic cosets of selforthogonal cyclic code, and then, prove the necessary and sufficient conditions of cyclic codes contain its Euclidean dual code and Hermitian dual code respectively. Finally, using the Calderbank-Shor-Steane(CSS) constructions and Steane’s constructions generate new families of quantum error correcting codes and quantum Maximum-Distance-Separable(MDS) codes, and these new quantum codes have parameters better than the ones available in the literature i.e. under the same code length, the dimensions of the new quantum codes is larger, or more quantum error can be corrected.2. We construct asymmetric quantum codes via classical BCH code and constacyclic code. We analyze the related properties of the cyclotomic cosets of selforthogonal cyclic code. Prove the necessary and sufficient conditions of cyclic codes contain its Euclidean dual code and Hermitian dual code respectively. Determine generator polynomial and definition set of cyclic codes, using the CSS constructions generate new families of asymmetric quantum codes and asymmetric quantum MDS codes. The examples show that these new quantum codes have parameters better and more various than the ones available in the literature.3. The product codes constructed by the repeated-root cyclic codes are used to construct the asymmetric quantum product codes. The parameters of asymmetric quantum product codes are also determined by the parameters of the repeated-root cyclic codes. The results show that these asymmetric quantum product codes based repeated-root cyclic codes are more efficient than some existent asymmetric quantum product codes, and this method can construct the good long asymmetric quantum codes with low decoding complexity. Finally, we simulate the performance of several examples of asymmetric codes constructed by the methods described in this paper. The results show that increasing channel asymmetry improves the performance of the asymmetric code.