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

JOURNAL OF COMBINATORIAL THEORY SERIES A

卷:119

Article

Katz, Daniel J.

关键词: Weil sum;    Binomial;    m-Sequence;    Cross-correlation;    Helleseth's conjecture;   

DOI  :  10.1016/j.jcta.2012.05.003

来源: Elsevier

【 摘 要 】

Let q be a power of a prime p, let psi(q) : F-q -> C be the canonical additive character psi q(x) = exp(2 pi iTr(Fq/Fp)(x)/p), let d be an integer with gcd(d, q - 1) = 1, and consider Weil sums of the form W-q,W-d(a) = Sigma(x is an element of Fq) psi(q)(x(d) + ax). We are interested in how many different values Wq,d(a) attains as a runs through Fq*. We show that if vertical bar{W-q,W-d(a): a is an element of F-q*}vertical bar = 3, then all the values in {W-q,W-d(a): a is an element of F-q*} 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 2(2 ''), then the cross-correlation cannot be three-valued. (C) 2012 Elsevier Inc. All rights reserved.

【 授权许可】

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