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Abstrakty
We consider Weil sums of binomials of the form
$W_{F,d}(a) = ∑_{x∈ F} ψ(x^{d} - ax)$,
where F is a finite field, ψ: F → ℂ is the canonical additive character, $gcd(d,|F^{×}|) = 1$, and $a ∈ F^{×}$. If we fix F and d, and examine the values of $W_{F,d}(a)$ as a runs through $F^{×}$, we always obtain at least three distinct values unless d is degenerate (a power of the characteristic of F modulo $|F^{×}|$). Choices of F and d for which we obtain only three values are quite rare and desirable in a wide variety of applications. We show that if F is a field of order 3ⁿ with n odd, and $d = 3^{r} + 2$ with 4r ≡ 1 (mod n), then $W_{F,d}(a)$ assumes only the three values 0 and $±3^{(n+1)/2}$. This proves the 2001 conjecture of Dobbertin, Helleseth, Kumar, and Martinsen. The proof employs diverse methods involving trilinear forms, counting points on curves via multiplicative character sums, divisibility properties of Gauss sums, and graph theory.
$W_{F,d}(a) = ∑_{x∈ F} ψ(x^{d} - ax)$,
where F is a finite field, ψ: F → ℂ is the canonical additive character, $gcd(d,|F^{×}|) = 1$, and $a ∈ F^{×}$. If we fix F and d, and examine the values of $W_{F,d}(a)$ as a runs through $F^{×}$, we always obtain at least three distinct values unless d is degenerate (a power of the characteristic of F modulo $|F^{×}|$). Choices of F and d for which we obtain only three values are quite rare and desirable in a wide variety of applications. We show that if F is a field of order 3ⁿ with n odd, and $d = 3^{r} + 2$ with 4r ≡ 1 (mod n), then $W_{F,d}(a)$ assumes only the three values 0 and $±3^{(n+1)/2}$. This proves the 2001 conjecture of Dobbertin, Helleseth, Kumar, and Martinsen. The proof employs diverse methods involving trilinear forms, counting points on curves via multiplicative character sums, divisibility properties of Gauss sums, and graph theory.
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Kategorie tematyczne
Czasopismo
Rocznik
Tom
Numer
Strony
181-199
Opis fizyczny
Daty
wydano
2015
Twórcy
autor
- Department of Mathematics, California State University, Northridge, 18111 Nordhoff St., Northridge, CA 91330-8313, U.S.A.
autor
- Institut de Mathématiques de Toulon, Université de Toulon, Avenue de l'Université, 83957 La Garde Cedex, France
Bibliografia
Typ dokumentu
Bibliografia
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Identyfikator YADDA
bwmeta1.element.bwnjournal-article-doi-10_4064-aa169-2-5
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