Construction Of Several Classes Of Minimal Linear Codes

Linear codes are an important part of cryptography and coding theory.Linear codes with low weight have important applications in multi-party security computing,data access systems,computer communication systems and other fields.With the development of The Times and the constant updating of computer and communication network,the research of linear code has made remarkable progress.Linear code has good reliability,which can effectively suppress the abnormal situation in the coding process,realize faster reading information and improve the reliability of the signal.The Hamming weight distribution of linear code can show its error-correcting ability and can be used to calculate the probability of information transmission error.Minimal linear codes are a special class of linear codes in which all nonzero codewords are minimal.Minimal linear codes have the structure of linear codes and can be used in many fields such as multi-user communication schemes,two-party security computing and secret key sharing.Therefore,it is important to construct different minimal linear codes,determine their parameters and calculate the weight distribution of minimal linear codes.Based on characteristic functions and generalized Boolean functions,this paper selects two methods to constructs several different classes of minimal linear codes.First,we choose true subspaces E₁,E₂ on F_q that satisfies E₁∩E₂=｛0｝,E₁^⊥∩E₂^⊥={0},let H=E₁^*∪E₂^*,f（x）=f_H（x）+Tr^m（ux）Tr^m（vx）,using the property of the characteristic function and the relationship between the vandermond determinant and the original identity root,the walsh transform of the function is calculated and the weight distribution of the first kind of linear codes is obtained according to the walsh transform.At the same time,some codes are selected from the first class of linear codes to construct minimal linear codes,and the minimality of this class of linear codes is proved.Then,we pick the definition set D,based on two special weak regular functions on F_q,a class of linear codes is constructed by using the special properties of functions,and the hamming weight distribution of the constructed linear codes are calculated,and the linear codes are proved to be minimal linear codes under certain conditions.Then a special subset of the definition set is selected to get a shorter class of puncture codes from the constructed linear codes and obtain the weight distribution of the puncture code.Under certain conditions,it is verified that the puncture code are minimal linear code.