Complete Weight Enumerators Of Linear Codes Over Galois Rings And Their Applications

The complete weight enumerators of linear codes can describe the internal structure of codes in detail by determining the frequency of the components in each codeword.By the complete weight enumerators of linear codes,the weight distribution of the codes can be obtained and the minimum distance of the codes can be determined.Few-weight linear codes have important applications in cryptography and combinatorial mathematics such as authentication codes,combination schemes,secret sharing schemes and constant combination codes.How to construct few-weight linear codes has always been an important content of coding theory.Determining the complete weight enumerators of linear codes is one of the commonly used and effective methods to construct few-weight linear codes.This paper mainly studies the complete weight enumerators of several types of linear codes determined by definition sets over Galois rings and their applications in cryptography and combinatorics.The thesis is organized as follows.In Chapter 1,we introduce the research background and significance of the complete weight enumerators of linear codes over Galois rings,the research status at home and abroad,and the summary of the main research results in this paper.In Chapter 2,we mainly introduce some common algebra,basic knowledge of coding theory,relevant results of exponential sum over finite fields and Galois rings,and basic concepts related to linear codes over finite fields and Galois rings.In Chapter 3,we construct two types of linear codes given by definition sets₁D over Galois rings,and accurately calculate the complete weight enumerators of these two types of linear codes by using the exponential sum and additive characteristics of finite fields and Galois rings.In Chapter 4,we construct two types of linear codes given by definition sets₂D over Galois rings,and accurately calculate the complete weight enumerators of these two types of linear codes by using the exponential sum and additive characteristics of finite fields and Galois rings.In Chapter 5,based on the complete weight enumerators of linear codes over Galois rings,we construct less-homogeneous weight linear codes over Galois rings.Using the Gray map from Galois rings to finite fields,we construct some Hamming few-weight linear codes over finite fields,and then study the application of these few-weight linear codes in authentication codes and constant combination codes.In Chapter 6,we summary the main results of this paper and list several open problems of error-correcting codes.