Construction Of Binary Linear Codes Based On Power Functions

The transmission of information is always carried out in a certain medium(such as wires,cables,fiber and radio waves).The physical characteristics of these media de-termine the inevitable errors in the transmission of information.When the information is wrong,it can automatically check errors and correct errors at the receiving end,the error correcting code has been widely used in channel coding.The linear code,based on its special characteristics,can find and correct errors in a greater probability and become the mainstream.At this point,the linear code has been applied in key shar-ing,authentication code,consumer electronics,communication industry,data storage system and so on.There are two main methods for constructing binary linear codes based on Boolean functions.The first method is to use Boolean functions to directly construct codewords in linear codes,that is,C(f)= {c=(Tr(af(x)+ bx))x?Fpm:a?Fpm,b E Fpm}.p is a prime number,and m is a positive integer,and this linear code can make up a linear code with a dimension of 2m at most.Another way is to use the subset of the finite field Fpm to construct a linear code:to take a set D={d1,d2,…,dn}(?)Fpm,then a linear code with a length of n can be constructedCD = {(Tr(xd1),Tr(xd2),…,Tr(xdn)):x?Fpm}.which the Tr is absolute trace function,and the set D is called the defining set of this code CD.The second method is widely concerned because of its flexible structure.Based on Bent functions,semi-Bent functions,almost Bent functions and quadratic functions,many good linear codes are generated.In this paper,we will focus on the binary case,and construct linear codes with few weights by choosing D as the preimage of the trace Trmn(·)of some power functions,i.e.,D ={x?F2n*|Trmn(fx))=0}where f(x)are power functions on F2n.We choose f(x)to be three special power functions:f(x)= x,f(x)= x2n-2i-1,f(x)= x2k+1.Some of the linear codes constructed through these functions are optimal in the sense that they meet the Plotkin bound on linear codes.