Study On The Construction Of Several Classes Of Linear Codes And Infinite Families 2-design

Linear codes with a few weights can be applied in secret sharing schemes,au-thentication codes,association schemes,data storage systems and combinatorics And design is an important concept in combinatorics,which can be used in coding theory,cryptography,communication and statistics.It is well known that there is a close relationship between coding theory and design.On one hand,a linear code over any finite field can be derived from the incidence matrix of a t-design.On the other hand,the supports of all codewords with a fixed weight in a linear and nonlinear code might hold a t-design under certain conditions.Constructing linear codes based on trace function is an important research direction in coding theory.If the trace representation is chosen properly,some optimized affine invariant codes can be obtained,afterwards 2-designs also could be determined.In this paper,we first construct several classes of linear codes with a few weights by use of trace representation,and obtain their weight distributions explicitly by utilizing number theory and exponential sums.We then prove that these codes are affine invariant by means of partial order theory.Thus,we derive infinite families of 2-designs from the supports of all the codewords with a fixed weight in these codes and the corre-sponding parameters.Finally,we illustrate the validity of our results using Magma program and it is further found that part of these linear codes can also be used to construct 3-designs.The main work is as follows(1).We obtain four classes of four-weight and six-weight linear codes derived from the p-ary cyclic codes with two non-zero.With the theory of character sums,the weight,distributions are discussed explicitly.And infinite families of 2-designs and their parameters are further studied(2).We design two classes of eight-weight and ten-weight linear codes from the p-ary cyclic codes with three non-zero and determine their weight distributions Then,seven and nine classes of infinite families of 2-designs and the corresponding parameters are obtained.Furthermore,it is also found that the two classes of linear codes can be used to support many 3-designs by Magma program(3).We construct a class of ten-weight and two classes of twelve-weight linear codes from non-binary Kasami cyclic codes,respectively.Then,the weight distri-butions and 2-designs are studied.