On Complete Weight Enumerators Of Cyclic Codes Over The Finite Chain Ring F_q+uF_q And Their Applications

With the development of computing technology and digital communication,the theory of error-correcting codes and cryptography have developed rapidly.One of the main objectives of error-correcting codes theory is to find good codes that can correct as many errors as possible and have little redundancy.These good codes should also have the characteristics of fast encoding and decoding.Cyclic codes are an important class of linear codes,which not only have good algebraic structure and quality,but also are easy to encode and decode.The complete weight enumerator not only describes the error correction ability of the code,but also helps to analyze and calculate the errors in the process of information transmission.It plays an important role in the engineering application background.In recent years,with the gradual improvement of complete weight enumerators over finite fields,coding scholars began to pay attention to complete weight enumerators and their application over finite rings.This thesis mainly studies complete weight enumerators of cyclic codes over the finite chain ring F_q+uF_q and their applications in cryptography.The thesis is organized as follows.In Chapter 1,we introduce the research background and significance,the overseas and domestic research status of cyclic codes and their complete weight enumerators over the finite chain ring F_q+uF_q.Moreover,the summary of main research contents of this thesis is introduced.In Chapter 2,we give some basic results of algebra and Gauss sums which lay the foundation for the subsequent writing of this thesis.In Chapter 3,according to the form of the elements in the finite chain ring F_q+uF_q,we divide the elements in the finite chain ring into five cases.Based on the definition of cyclic codes and properties of trace mapping over the finite chain ring F_q+uF_q,using the Gaussian sums over finite fields,we determine the complete weight enumerators of any codeword in each case,and then the complete weight enumerators of cyclic codes are determined.In Chapter 4,based on the complete weight enumerators of the cyclic codes over the finite chain ring F_q+uF_q,the few-weight linear codes over the finite field Fq are constructed by using the Gray mapping from finite chain ring F_q+uF_q to finite field Fq,and then the application of few-weight linear codes in authentication codes,minimum codes,constant weight linear codes and constant composition codes is studied.In Chapter 5,we summary the main results of this thesis and list several open problems of complete weight enumerators.