Research On Constacyclic Codes And Their Applications In Information Security

The theory of error-correcting codes is not only the theoretical basis of information security, but also the theoretical basis of quantum information. Classical error-correcting codes over finite fields have become mature in theory, and attained extensive applications in practice. With a more and more deeply study in the theory of error-correcting codes, the theoretical value and practical significance of error-correcting codes over finite rings have gradually been recognized. Research on the theory of error-correcting codes over finite rings have been a hot spot recent years. During these studies, constacyclic codes (including cyclic codes and negacyclic codes) over finite rings have become the keystone in the study of error-correcting codes over finite rings. In 1998, Calderbank et al. established the mathematical theory of quantum error-correcting codes, and gave an effective way to construct quantum error-correcting codes by using error-correcting codes, which has enormously motivated the application of error-correcting codes in quantum information.In this dissertation, the theory of constacyclic codes over finite rings is taken as the foundation, while the construction of quantum error-correcting codes is taken as the application. In the theory of constacyclic codes over finite rings:Firstly, we studied negacyclic codes of length 2ps over Fpm+uFpm, where u2= 0. Negacyclic codes of such length are classified and the detailed structures are provided. Among other results, the number of codewords, and the dual of each negacyclic code are obtained and the necessary and sufficient conditions for the existence of negacyclic self-dual codes are also given. Secondly, a new Gray map is defined from F2+uF2+u2F2+u3F2, to F2 with u4=0. It is proved that the Gray image of a linear (1+u+u2+u3)-constacyclic code of length n over such ring is a distance-invariant linear cyclic code of length 4n over F2. Further more, the generator polynomials of such constacyclic codes under the Gray map is determined, and some optimal binary linear cyclic codes are also obtained. Finally, we studied (1+u+v)-constacyclic codes of length 2s over F2+uF2+vF2+ uvF2, where u2=v2=0,uv=vu. Constacyclic codes of such type and such length are classified and the detailed structures are provided. Among other results, the number of codewords, and the dual of each constacyclic code are obtained due to the classification. The necessary and sufficient conditions for the existence of constacyclic self-dual codes and the enumeration of constacyclic self-dual codes are also given. In the construction of quantum error-correcting codes:Firstly, two new classes of quantum MDS codes with general large minimum distance are constructed through constacyclic codes over Fq2 by using Hermitian construction. Secondly, three classes of quantum repeated-root cyclic codes with optimal parameters are constructed through repeated-root cyclic codes of length 2ps over Fq by using Steane’s enlargement construction. Thirdly, six new classes of optimal asymmetric quantum codes with great asymmetric are constructed through constacyclic codes over Fq2 by using CSS construction. Finally, four new classes of optimal quantum convolutional codes are constructed through constacyclic codes over Fq2 by using Piret construction and the construction method introduced by La Guardia.