Research On Two Classes Of Linear Codes Over Finite Fields And Their Applications

The research on error-correcting codes theory over finite fields has been well studied and widely used in various communication systems and computer systems.However,there are still many problems need to be solved and further developed.At the end of the 20 th century,quantum error-correcting(QEC)codes was born in order to ensure the realization of quantum computing and quantum communication.In 1998,Calderbank et al.presented the mathematical form of QEC codes and proposed a systematic and effective mathematical method for constructing QEC codes.This had greatly stimulated researchers' enthusiasm,making the constructions of QEC codes become a hot research topic.With further study on the theory of QEC codes,quantum synchronizable(QS)codes and entanglement-assisted quantum error-correcting(EAQEC)codes were proposed as two new types of QEC codes and received attention by many scholars.Cyclic codes and generalized Reed-Solomon codes are two classes of important linear codes over finite fields.In this dissertation,three classes of optimal ternary cyclic codes are constructed and their weight distributions are determined completely.Secondly,the weight distributions of a class of p-ary cyclic codes with three non-zeros for all distinct cases are determined explicitly,and their applications in secret sharing schemes is introduced.In addition,two classes of dual-containing cyclic codes are obtained by using the cyclotomy of order four,and two new classes of quantum synchronization codes are constructed based on these codes and their expansion codes.Finally,based on the generator(or parity check)matrices and Hermitian hulls of generalized Reed-Solomon codes,several classes of EAQEC codes are constructed.Specific research contents are as below:1)We prove that three classes of ternary cyclic codes are optimal,which means that their parameters meet some certain bound.The weight distributions of their dual codes are also determined completely.The results indicate that their dual codes have few non-zero weights.Some of their duals are also optimal codes.2)The weight distributions of a class of p-ary cyclic codes for all distinct cases are determined explicitly.Some of these p-ary cyclic codes are shown to be the best-known codes by specific examples.Moreover,the covering structures of the class of p-ary cyclic codes are studied and used to construct secret sharing schemes.3)Some cyclic codes are constructed based on the cyclotomy of order four.Furthermore,two classes of QS codes with good parameters are constructed based on the constructed cyclic codes.These constructed QS codes are Calderbank-Shor-Steane(CSS)QEC codes that can tolerate maximum number of misalignment errors.In addition,these QS codes usually possess good error-correcting capability towards bit error and phase error,due to that many cyclic codes employed for construction are optimal or almost optimal.4)Two classes of EAQEC MDS codes are constructed by studying the generator matrix and parity check matrix of generalized Reed-Solomon codes over finite fields.By extending the two conclusions,we obtain four new classes of EAQEC codes.Among those,most are new in the sence that the parameters of EAQEC codes are different from all known ones.The EAQEC MDS codes constructed have much larger minimum distance than the known QEC MDS codes.In particular,some of our EAQEC MDS codes have much larger minimum distance than the known codes with same length and same number of ebits.Finally,we constructed three classes of EAQEC codes and three classes of EAQEC MDS codes by studying the dimensions of the Hermitian hulls of genneralized Reed-Solomon codes over finite fields.Compared with known EAQEC codes,these codes constructed are new and their maximally entangled states can take various values.Moreover,these EAQEC codes have more flexible lengths.