Research On Quantum Error-Correcting Codes In Information Security

Since the late 20th century,quantum computation and quantum communication have become a crossover and frontier subject of computer science,communications,mathematics and physics.As the classic digital communication,in order to realize quantum computation and quantum communication,then must solve the problem of quantum error correction.In 1996,Calderbank,Shor and Steane independently came up with a systematic and effective method about how to use mathematical tools to construct quantum error correcting codes at the same time,and set up a bridge between classical error correcting codes and quantum error correcting codes.This greatly promotes the rapid development of quantum error correcting codes.Since then,scholars begin to do a deep research in the theory of quantum error correcting codes.As a result,in the research process and the practice of communication,the theory of quantum error correcting codes has been improved continuously,and has formed many branches,such as asymmetric quantum codes,entanglement-assisted quantum codes,quantum convolutional codes and so on.In this paper,we mainly use constacyclic codes over finite fields and cyclic codes over finite rings as the theoretical basis,and make a deep research on the theory of quantum error correcting codes.Firstly,when q=2e,e>1 is odd,and the length of code is n=(q2+1)/5 over finite fields,the q2-cyclotomic coset Ci of the module(q+1)n has been given by calculation.Further,the necessary and sufficient condition of ?-constacyclic codes with length n=(q2+1)/5 over finite fields contains its Hermitian dual codes is obtained,while q=2e,e ? 1mod4.On this basis,we construct a class of asymmetric quantum codes.According to the Singleton bound of asymmetric quantum codes,this kind of asymmetric quantum codes is the optimal.Similarly,we also give the necessary and sufficient condition of ?-constacyclic codes with length n=(q2+1)/5 over finite fields contains its Hermitian dual codes,when q=2e,e?3mod 4.Then,we construct the other family of asymmetric quantum codes.Besides,the second kind of asymmetric quantum codes is also the optimal by the Singleton bound of asymmetric quantum codes.Secondly,the definition of decomposing the defining set of constacyclic codes is given,and prove that if the defining set Z is decomposed into Z=Z1 ? Z2,then the entangled state is c=|Z1|.Thus,we solve the problem of determining the number of entangled state c.Through decomposing the defining set of constacyclic codes C,we obtain the entangled state c of those four selected cases for a certain length n=(q2+1)/5 successfully.Then,by using the construction method of entanglement-assisted quantum codes,four types of entanglement-assisted quantum codes are constructed.According to the Singleton bound of entanglement-assisted quantum codes,those four types of entanglement-assisted quantum codes are optimal.Besides,through decomposing the defining set of negacyclic BCH codes C over finite fields and determining elements number of two cyclotomic cosets for a certain length n=(q4m-1)/(q2-1),we get the number of entangled state c for the selected length.Then,we obtain its dimension by calculation.Next,by using the construction method of entanglement-assisted quantum codes,we construct a family of entanglement-assisted quantum codes with larger length and dimension.Finally,the structure of cyclic codes over the finite ring R=F2m+uF2m+vF2m+uvF2m with odd length is studied,and the self-orthogonal codes in finite fields are obtained by defining a Gray map.In the end,some quantum error correcting codes with new parameters are constructed by using Euclidean self-orthogonal cyclic codes over finite rings.