Weil sums of binomials, three-level cross-correlation, and a conjecture of Helleseth
type: Item
instance of: scholarly article
title: Weil sums of binomials, three-level cross-correlation, and a conjecture of Helleseth
zbMATH Open document ID: 1263.11111
author: Daniel J. Katz
published in: Journal of Combinatorial Theory. Series A
publication date: 2012-08-24T00:00:00Z
Mathematics Subject Classification ID: 11T24
Mathematics Subject Classification ID: 11T71
zbMATH DE Number: 6071600
zbMATH Keywords: Weil sum
zbMATH Keywords: Helleseth's conjectures
zbMATH Keywords: Walsh spectrum
zbMATH Keywords: cross-correlation
Wikidata QID: Q123265522
MaRDI profile type: Publication
full work available at URL: j.jcta.2012.05.003
OpenAlex ID: W2037565534
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DOI: 10.1016/J.JCTA.2012.05.003
review text: Let \(q\) be a power of a prime \(p\) and let \(\psi_q: \mathbb F_q\to\mathbb C\) be given by \(\psi_q(x)=\exp (2\pi iTr(x)/p)\). For an integer \(d\) prime to \(q-1\), the Weil sums studied here are: NEWLINE\[NEWLINE W_{q,d}(a)=\sum_{x\in\mathbb F_q} \psi_q(x^d+ax). NEWLINE\]NEWLINE The author considers two long-standing conjectures of Helleseth:NEWLINENEWLINE(A) If \(q>2\) and \(d\equiv 1\pmod{p-1}\) then \(W_{q,d}(a)=0\) for some \(a\in\mathbb F_q^*\).NEWLINENEWLINE(B) If \(q=p^{2^n}\) and \(d\) is non-degenerate then \(W_{q,d}\) takes at least four values over \(\mathbb F_q^*\).NEWLINENEWLINEThe main theorem here is: If \(W_{q,d}\) is 3-valued over \(\mathbb F_q^*\) then \(d\equiv 1\pmod{p-1}\) and \(W_{q,d}(a)\in\mathbb Z\) for all \(a\in \mathbb F_q\). This implies that, for fixed \(q\) and \(d\), at least one of (A) or (B) must hold. It is further shown that conjecture (B) holds for \(p=2\). A helpful appendix gives equivalent formulations in information theory (Walsh spectrum), cryptography (cross-correlations) and finite geometry (hyperplane intersections).
description: scientific article; zbMATH DE number 6071600