Applications Of Exponential Sums In Cycle Codes

In coding theory,the error correcting capability of cyclic codes depends on the minimum non-zero weight of cyclic codes,so the weight distribution of cyclic codes is of great research value.However,the weight distribution of cyclic codes is generally a difficult problem,because in fact,it is determined by the corresponding exponential sums and the value distribution.So far,the weight distribution of most cyclic codes is still unknown.Exponential sum is an important branch of number theory.In this thesis,we mainly study the exponential sums and its applications in cyclic codes and linear codes,including the weight distribution of cyclic codes and linear codes.The main work of this paper is as follows:1)Let F be a finite field with q elements,q = pm,? is a primitive element of Fq.We mainly investigate the weight distribution of cyclic codes whose duals have two zeros of same orders.Suppose that h(x)is a reducible polynomial whose roots are ?-1 and ?-d,d=(pm-1)(pk+3)+2(pk+1)/4.We construct cyclic code C[n,2m]with h(x)as parity-check polynomial,the expression is:where c(?,?)=(c0,c1,…,cn-1),(?,? ?Fq)and ci =Trptq(??id + ??i)? Fq0,0?i?n-1 According to the knowledge of coding theory,we give the weight expression of the codeword and the problem is transformed into the weight distribution of T(?,?),We let x= y2 and x ??y2,? is a primitive element of Fq0,in this way,we can derive the exponential sum distribution of the quadratic formThrough a series of calculations,the function T(?,?)has three values,±p(m+t)/2,0.Finally,the weight distribution of cyclic code is obtained,we will obtain a class of five weight cyclic codes and a class of three weight cyclic codes.2)Our second work is to construct binary linear codes C by means of defined sets,let q = 2m,l =(m,h),our defined sets isDefine a linear code of length n over GF(2)byAccording to the definition of weight distribution,we derive a formula for exponential sum,and eventually obtain a three weight linear code and its exact weight distribution.