Studies On The Coding Theory Based On Algebraic Methods And Applications

Coding theory is the theoretical basis of information security. The coding theory over finite fields has become relatively perfect and it has been applied to practice. Recently, coding theory over finite ring has attracted many scholars and scholars have paid close attention to cryptography based on coding theory, meanwhile, coding theory applied to quantum information is also a subject of quantum information and quantum computing.A lot of scholars utilize self-orthogonal classical codes to construct quantum codes with good parameters. How to construct quantum codes with good parameters have become a hot topic that attracted some coding and cryptography experts. In the past nearly twenty years, primitive non-sense-narrow BCH cyclic codes have been applied to construct quantum codes with good parameters. Nonprimitive non-sense-narrow BCH cyclic codes and negacyclic codes have attracted a few of scholars and these codes are applied to construct quantum codes with good parameters until the past two or three years, meanwhile, quantum codes with good parameters constructed from nonprimitive non-sense-narrow BCH cyclic codes and negacyclic codes have become an important method in quantum information and quantum computing. Constacyclic codes contain cyclic codes and negacyclic codes, and quantum codes with good parameters constructed from constacyclic codes is also a hot topic that has attracted some coding and cryptography experts in the past two or three years.In this thesis, we study the coding theory over finite rings and coding theory are applied to cryptography and quantum information. The details of this dissertation are given as follows:1. M-spotty Hamming weight enumerator of linear codes over finite ring Fp+vFp(v2= v) are studied and we obtain MacWilliams type identity for m-spotty Hamming weight enumerator of linear codes over finite ring Fp+ vFp(v2= v). Moreover, m-spotty Rosenbloom-Tsfasman weight enumerator of linear codes over finite ring Fq[u]/(uk)(uk=0) is also studied and we obtain MacWilliams type identity for m-spotty RosenbloomTsfasman weight enumerator of linear codes over finite ring Fq[u]/(uk)(uk= 0).2. The method of constructing MacDonald over finite ring Fq[u]/(us)(us= 0) is studied, meanwhile, we provide Hamming distribution of torsion codes constructed from MacDonald codes. Moreover, we use torsion codes constructed from MacDonald codes to determine minimum codes and obtain a class of access structure of secret sharing scheme.3. The properties of cyclotomic cosets of nonprimitive non-sense-narrow BCH cyclic codes are used to construct some asymmetric quantum codes with good parameters, and we use cyclotomic cosets to character a class of nonprimitive non-sense-narrow BCH codes. Moreover, nonprimitive non-sense-narrow BCH cyclic codes are applied to construct some families of quantum convolutional codes with good parameters.4. We study the constructions of quantum error-correcting codes from negacyclic codes. Fistly, we use cyclotomic cosets of negacyclic codes to construct two families of optimal asymmetric quantum codes. But a lot of optimal asymmetric quantum codes are always constructed based on cyclic codes before we use negacyclic codes to construct optimal asymmetric quantum codes. Secondly, we study the structure of quantum convolutional codes and we use some families of negacyclic codes to construct quantum convolutional codes with good parameters. Cyclic codes are utilized to construct quantum convolutional codes before a few of scholars use negacyclic codes to construct quantum convolutional codes. Finally, we study the relationship among negacyclic codes, quantum subsystem codes and entanglement-assisted quantum codes, using negacyclic codes to construct a family of optimal quantum subsystem codes and three families of entanglement-assisted quantum codes that satisfy the entanglement-assisted Singleton bound.5. We study the constructions of quantum error-correcting codes from contacyclic codes. Firstly, we use constacyclic codes to construct some families of optimal asymmetric quantum codes. And some classes of quantum convolutional codes with good parameters, where some quantum convolutional codes constructed here are optimal. Secondly, some extensions of the results of constacyclic codes are studied, and we use these results to construct some families of quantum convolutional codes with good parameters. Finally, the relationship between constacyclic codes and entanglement-assisted quantum codes is studied, and we use constacyclic codes to construct some families of entanglement-assisted quantum codes with good parameters.