Cyclotomy of Weil Sums of Binomials

2013/12/13 by Aubry, Yves, Katz, Daniel J., Langevin, Philippe
#11L05 #11T22 #11T23 #Combinatorics (math.CO) #FOS: Computer and information sciences #FOS: Mathematics #Information Theory (cs.IT) #Number Theory (math.NT)

paper · doi:10.48550/arxiv.1312.3889

Abstract

The Weil sum W_K,d(a)=∑_{x ∈ K} ψ(x^d + a x) where K is a finite field, ψ is an additive character of K, d is coprime to |K^×|, and a ∈ K^× arises often in number-theoretic calculations, and in applications to finite geometry, cryptography, digital sequence design, and coding theory. Researchers are especially interested in the case where W_K,d(a) assumes three distinct values as a runs through K^×. A Galois-theoretic approach, combined with p-divisibility results on Gauss sums, is used here to prove a variety of new results that constrain which fields K and exponents d support three-valued Weil sums, and restrict the values that such Weil sums may assume.

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• Proof of a Conjectured Three-Valued Family of Weil Sums of Binomials
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