On the Values and Spectrum of Weil Sum of Binomials

Abstract: The Weil sum of an additive character $\mu$ over a finite field $F$ is defined to be $W_{F,s}(a)=\sum_{x \in F} \mu(x^s-ax)$ where $s$ is an integer coprime to $|F^*|$. The Weil spectrum counts distinct values of the Weil sum as $a$ runs through the invertible elements in the finite field. Determining the values of these sums and the size of its spectrum give answers to long-standing problems in cryptography, coding and information theory. In this talk, we prove a special case of the Vanishing Conjecture of Helleseth ($1971$) on the presence of zero in the Weil spectrum. We then propose a new conjecture on when the Weil spectrum contains at least five elements, and prove it for a certain class of Weil sum.

number theory

Audience: researchers in the discipline

Comments: To join the talks on January 19th, please register here:

ucsd.zoom.us/meeting/register/tJIuf-6uqjoiH9eVbtLN9y1G9l0qEBmbDRmV

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POINT: New Developments in Number Theory

Series comments: There will be two 20-minute contributed talks during each meeting aimed at a general number theory audience, including graduate students.

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