Studying On Extensions Of MDS Codes And Related Problems

Information theory is a kind of subject developed from communication system.The early information theory is mainly used for studying the problems of information source,sink,transmission and encoding.In 1948,Shannon published "A mathematical Theory of Communication",which laid the foundation of information theory.Since then,Shannon published "Communication Theory of Secrecy System",which laid the foundation to the study of modern cryptology.According to different encoding purposes,codes can be divid-ed into source codes,channel codes,and secret codes.Codes with the function of error-correcting in channel coding are called error-correcting codes.Error-correcting codes are a class of important codes.They have important applications in the field of communication.Maximum distance separable codes(MDS codes)are an important kind of error-correcting codes.They have many good properties.For example:the weight distribution of MDS codes is completely determined,any k position can be used as the information bits;Many good codes are MDS codes,such as:Reed-Solomon codes.MDS codes also have some good applications,for example:constructing erasure codes.Self-dual codes are also a kind of error-correcting codes with many good properties.As a result,a large number of scholars have studied a variety of self-dual codes,such as:MDS Euclidean self-dual codes,MDS Hermitian self-dual codes.And Hermitian self-dual codes have important applica-tions,for example:constructing erasure codes.In this thesis,we review some theoretical knowledge of linear codes,MDS codes and near MDS codes.Then,we introduce the construction of MDS Euclidean self-dual codes by equation,and we improve the method.Many new MDS Euclidean self-dual codes with good parameters are obtained from our constructions.Meanwhile,We obtain some near MDS Euclidean self-dual codes by studying the splitting.In Chapter 4,we introduce the construction of MDS Euclidean self-dual codes by Reed-Solomon codes.Also we extend the construction method and obtain many new MDS Hermitian self-dual codes with good parameters.Meanwhile,we correct Theorem 8 in[26].And we obtain some near MDS Hermitian self-dual codes by studying the Ⅰ,Ⅱ splitting.At last,we end this thesis with a summary and prospect.