Four Classes Of Linear Codes And Their Weight Distribution Based On Quadratic Form

With the continuous improvement of social information level,a large amount of information such as communication information,funds transfer,electronic medi-cal records are exchanged through computer network and information channel.Errors will occur in the process of transmission if message interfered by noise.Error detection and correction can be achieved by error correcting code theory.Linear codes have good algebraic structure and are easy to describe,encrypt and decrypt.Its weight distribution provides a lot of information in the possibility of error detection and correction.There-fore,it is an important topic in the theory of error correcting codes to select a suitable definition set to construct linear codes and study their weight distribution.In this paper,linear codes are constructed by using the preimage set of quadratic function as the definition set.Through a large number of magma experiments,the data are collected and sorted out,and the rules of data are found and summarized.We obtained four classes of linear codes with low weight having the form as in CD={(Tr1m(aα1d),Tr1m(aα2d),…,Tr1m(aαnd))|a∈F2m},D={x∈F2m*|Tr1m(xr)=0}={α1,α2,…,αn}by simplifying the formula,the problem of determining the weight distribution of lin-ear codes is transformed into solving exponential sums.Using quadratic type theory,to solve exponential sums,the rank of quadratic type needs to be determined first.Ex-clude impossible values when possible values of an exponential sum are found,and the possible values of an exponential sum are found,the impossible values are excluded,and the weight are determined by substituting it into calculation.Finally,the weight distribution of linear codes is determined by pless power moment formula.