The Expolring Of The Binary Optimal Code Of Ring Code Based On The Gray Mapping

Claude Shannon published the article about Mathematical Theory of Communication in 1848,and pointed out the existence of the error correcting code.After that,the error correcting code has been rapid development.E.Prange first introduced the concept of linear code and cyclic code in 1957.He had made the promotion of the research into the ring,And it caused a profound impact on the remaining ring.The code of the ring attracted more and more attention since 1990,including the code of Z₄ ?Z_2^k ?Z₂Z₄-?Z₂Z_2^k .In addition,the binary code of Gray Mapping also made a major breakthrough.These codes of the ring already have shown their good prospects in the practical application.This paper is to study the nature of the code based on the previous studies,and discuss the binary code of Gray Mapping.And illustrated by an example,this additive group at Gray mapping can also get good binary linear code.This article is divided into four sections to explore the ring code in binary optimal code of the Gray Mapping.?1?Introduction: We Summary the content of coding theory,and introducethe error correcting code.Commonly used error correcting codes aregiven,such as linear codes,cyclic codes,etc.Then we define theconcept and property of these codes.?2?We studied the code on the ring,including the structure of the code andcyclic codes.Then we elaborathe binary code under the Gray Mapping,and We got the optimal parameters which could to be a good code bycalculating.?3?We studied the Z₂Z_2^k code,including the structure of Z₂Z₄-codeandZ₂Z_2^k code?the binary code under the Gray Mapping,and We got theoptimal parameters which could to be a good code by calculating.?4?On the basis of the first three parts of the study,We promote the ringcode onto theZ₂ R?R=F₂+uF₂?code.Then we studied the structure ofthe ring and the nature of the code.After that,The the binary code underthe Gray Mapping were studied in this part.Finally we completed theexpolring of the binary optimal ring code under the Gray Mapping,andgot the optimal parameters which could be a good code.