The Weight Distribution Of A Class Of Codes Over GF(2)

Let r = 2ⁿ,q = 2^k, where n and k are integers, k|n. If n is even,write n = 2n₀, where n0is odd. Let GF(r) be a finite field with r elements, where r is a power of a prime. In this paper,we define a special class of cyclic codes C = {c(a,b)|a,b∈GF(r)}, wherec(a,b) = (Tr_r/q(a(β⁰)(2^α+1) + bβ⁰),Tr_r/q(a(β¹)(2^α+1) + bβ¹),···,Tr_r/q(a(β^2n-2)(2^α+1) + bβ^2n-2) ,Tr_r/q is the trace mapping from GF(r) to GF(q),βis a generator of GF(r)*, r = 2ⁿ,q = 2^k,n and k are positive integers and k|n. If n is even and n = 2n₀, then we assume n₀ is odd. Wedetermine the value of the exponential sum∑x∈GF(r)χ(ax(2^α+1) + bx) (a,b∈GF(r)),whereαis even,χ(x) = (-1)^Tr(x) is the canonical additive character of GF(r), Tr is the tracemapping from GF(r) to GF(2). Thus the weight distribution of C is obtained when k = 1. Inaddition, we prove the trace expression of cyclic codes by matrices skill.