Weil sums of binomials, three-level cross-correlation, and a conjecture of Helleseth

摘要

Let q be a power of a prime p, let ψq:Fq→C be the canonical additive character ψq(x)=exp(2πiTrFq/Fp(x)/p), let d be an integer with gcd(d, q-1)=1, and consider Weil sums of the form W_(q,d)(a)=∑x∈Fqψq(xd+ax). We are interested in how many different values W q,d(a) attains as a runs through Fq*. We show that if |{Wq,d(a):a∈Fq~*}|=3, then all the values in {Wq,d(a):a∈Fq~*} are rational integers and one of these values is 0. This translates into a result on the cross-correlation of a pair of p-ary maximum length linear recursive sequences of period q-1, where one sequence is the decimation of the other by d: if the cross-correlation is three-valued, then all the values are in Z and one of them is -1. We then use this to prove the binary case of a conjecture of Helleseth, which states that if q is of the form 22n, then the cross-correlation cannot be three-valued.