New Constructions Of Self-dual Codes And Comple-Mentary-dual Codes Via(Extend)Generalized R-S Codes

Error-correcting codes are an imporant topic in coding theory since it can lower the probability of error,and improve the reliability of communication system.Combinatorics?elementary number theory?linear algebra?modern algebra are the important tools in the study of coding theory.In recent twenty years,more and more abstruse mathematical tools such as algebra geometry?algebra number theory?group representation theory are applied in the study coding theory.MDS codes are a class of special optimal linear codes due to it reaches the Singleton bound,it is an important topic in coding theory since it has high ability to correct error.There are close relation between the Self-dual MDS codes?linear complementary dual MDS codes and combinatorial design?lattice?secret sharing?invariant theory?quantum correcting codes?In recent years,the study of Self-dual MDS codes and linear complementary dual MDS codes have attracted a lot of attention from scholars.A basic topic in error-correcting theory is to construct error-correcting codes with good property over finite fields(or rings).There are many different constructions of Self-dual MDS codes and linear complementary dual MDS codes,we classified these constructions into the following three types based on the tools use in the constructions:(1)the constructions based on coding theory;(2)the constructions based on combinatorics;(3)the constructions based on algebra.In this thesis,we express five classes of the constructions of Self-dual MDS codes and two classes of the constructions of linear complementary dual MDS codes,based on the(extend)generalized Reed-Solomon codes,and also applied the coding theory?finite fields and the theory of modern algebra,these construction will enrich the mathematical theory of error-correcting codes.