Research On Constructing Minimal Linear Codes With Vectors From Given Weights

Linear codes have important applications in consumer electronics,data storage,communications,cryptography and combinatorics.Minimal linear codes are a special kind of linear codes.Minimal linear codes can be used in both secret sharing schemes and secure two-party computation.The construction of minimal linear codes has far-reaching practical significance.How to construct and characterize minimal linear codes with good performance is one of the hotspots in recent years.In this paper,we define two special sets,discuss weights of solutions of a system of linear equations,and construct and characterize minimal linear codes from vectors with given weights.Finally,we present some results in previous papers,which are contained in this paper.The structure of this paper is as follows:The first chapter is the introduction,which describes the background of coding theory.Then it introduces the development of coding theory and the research direction of coding theory.Finally,it briefly explains the main results of this paper.The second chapter expounds the basic theory,which includes mathematical theory of error correcting codes,algebra theory and the related results of minimal linear codes.These theoretical results lay the foundation for the construction of the following minimal linear codes.The third chapter is the main content of this paper.This chapter first defines two special sets and discusses weights of the solutions of a system of linear equations.Then,we construct and characterize minimal linear codes from vectors with given weights.Finally,we present some results in previous papers,which are contained in this paper.The fourth chapter concludes the work of this paper,and presents some further directions of minimal linear codes.