Constructions And Applications Of Optimal Algebraic Codes

As a branch of mathematics and computer science,coding theory has been de-veloped more than seventy years.With the efforts of researchers,coding theory has become an independent subject with valuable applications.Maximum distance separa-ble(MDS)code is a kind of optimal algebraic codes which has optimal error-correcting ability.Therefore,the construction of MDS codes is one of the main topics of algebraic coding.In this thesis,we study some constructions of MDS codes,include MDS Eu-clidean self-dual codes,MDS Euclidean self-orthogonal codes,MDS Euclidean almost self-dual codes and the intersection of MDS codes.Moreover,as an application of op-timal algebraic coding in quantum coding,we discuss entanglement-assisted quantum error-correcting codes in the asymmetric quantum channels with maximum distance separability.In Chapter 1,we introduce the history and background of coding theory and quan-tum codes,and we summarize our contributions of this thesis.In Chapter 2,we consider the constructions of MDS self-dual codes and other optimal algebraic codes.We give four new families of MDS Euclidean self-dual codes with flexible parameters.And we make a comparison of our results with the previous results and state that our constructions can take up a proportion of generally more than34%.Moreover,we give two new families of MDS Euclidean self-orthogonal codes and MDS Euclidean almost self-dual codes.In Chapter 3,we make a research of intersection of MDS codes.As a complemen-tary constructions of the previous results,we give linearl-intersection pairs of MD-S codes over Fqfor length n=q or q+1.Moreover,we give all possible linearl-intersection pairs of MDS codes over F2m for length n=2~m+2.In summary,all the possible linearl-intersection pairs of MDS codes are given.In Chapter 4,we consider the topic about asymmetric entanglement-assisted quan-tum error-correcting codes(AEAQECCs).We introduce the information of quantum codes,entanglement-assisted quantum error-correcting codes(EAQECCs),asymmet-ric quantum error-correcting codes(AQECCs)and AEAQECCs.Then we give the notions about CSS-type AEAQECCs.Finally,we construct all the possible pure MDS CSS-type AEAQECCs.In Chapter 5,we make a conclusion of this thesis and present some valuable prob-lems in the future work.