The Construction Of Several Classes Minimal Linear Code Over Finite Fields

A linear codes of length n over the finite F_q is simply a subspace of the vector space F_qⁿ.Linear codes are widely used in computer systems,communication systems,data storage,information security,digital signatures,multi-party secure computing,military,satellite communications and other fields.Very small linear codes are used as A kind of special linear codes with good algebraic structure and properties,which are widely used in secret sharing and data communication and storage.This paper uses the function constructed by the trace function to study the structure of this type of linear code,and obtains the Hamming weight distribution of this type of linear code.This method of constructing a minimal linear code relies on the calculation of the elements on the finite field.The value under Walsh transformation of a class function.In order to calculate the distribution of this kind of Walsh transformation,this paper gives an algebraic equation between the van dermond determinant and the subprimitive unit root,so that we can get a kind of trace The relationship between the Walsh transform of the function formed by the function and this Vandermonde determinant and its algebraic remainder.Using this relationship,determine the specific function constructed by this type of trace function,and obtain the Hamming weight distribution of a series of linear codes.Select some code words in this type of linear code,and verify that the selected code is a very small linear code combined with the method of classification discussion.This article is divided into four chapters,namely:The first section of the first chapter mainly discusses the research background of minimal linear codes,the research status at home and abroad and the work of this paper,as well as the advantages and disadvantages of the minimal linear codes constructed in this paper compared to other minimal linear codes.The second chapter mainly discusses the relevant algebraic background of this article.The first section introduces the related properties of finite fields,Galois groups and trace mapping.The second section introduces the concept of minimal linear codes and the calculation of a special function structure The related lemmas of the Hamming weight distribution of linear codes,and the Walsh transformation of the characteristic function of any non-empty subset on a finite field in a specific function form.The third chapter mainly discusses the construction of linear minimal codes on finite fields.The first section introduces the construction of minimal linear codes using the characteristic functions of subsets on finite fields.The second section introduces the use of non-empty linear subspaces on finite fields.The characteristic function of constructs a minimal linear code.Chapter 4 mainly introduces the structure of linear code when the function is f(x)=x^pm-1+Tr(ux)Tr(vx).