Divisibility of Weil Sums of Binomials

2014/07/30 by Katz, Daniel J.
#11L05 #11L07 #11T23 #11T71 #Combinatorics (math.CO) #FOS: Computer and information sciences #FOS: Mathematics #Information Theory (cs.IT) #Number Theory (math.NT)

paper · doi:10.48550/arxiv.1407.7923

Abstract

Consider the Weil sum W_F,d(u)=∑_{x ∈ F} ψ(x^d+u x), where F is a finite field of characteristic p, ψ is the canonical additive character of F, d is coprime to |F^*|, and u ∈ F^*. We say that W_F,d(u) is three-valued when it assumes precisely three distinct values as u runs through F^*: this is the minimum number of distinct values in the nondegenerate case, and three-valued W_F,d are rare and desirable. When W_F,d is three-valued, we give a lower bound on the p-adic valuation of the values. This enables us to prove the characteristic 3 case of a 1976 conjecture of Helleseth: when p=3 and [F:\mathbb F₃] is a power of 2, we show that W_F,d cannot be three-valued.

Related

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