On Weil Sums, Conjectures of Helleseth, and Niho Exponents

TL;DR

This paper investigates Weil sums over finite fields, focusing on Niho exponents, providing new proofs for existing conjectures, and proposing new conjectures related to the spectrum of these sums.

Contribution

It offers a new proof of Helleseth's Vanishing Conjecture for Niho exponents and introduces a conjecture about the size of the Weil spectrum, proving it in specific cases.

Findings

Confirmed the presence of zero in the Weil spectrum for Niho exponents.

Provided bounds and spectral analysis of Weil sums in the Niho setting.

Proposed and proved a conjecture on the minimum size of the Weil spectrum in certain cases.

Abstract

Let $F$ be a finite field, $\mu$ be a fixed additive character and $s$ be an integer coprime to $|F^\times|$. For any $a\in F$, the corresponding Weil sum is defined to be $W_{F,s}(a)=\displaystyle\sum_{x \in F} \mu(x^s-ax)$. The Weil spectrum counts distinct values of the Weil sum as $a$ runs through the invertible elements in the finite field. The value of these sums and the size of the Weil spectrum are of particular interest, as they link problems in coding and information theory to other areas of math such as number theory and arithmetic geometry. In the setting of Niho exponents, we examine the Weil sum, its bounds and its spectrum. As a consequence, we give a new proof to the Vanishing Conjecture of Helleseth ($1971$) on the presence of zero in the Weil spectrum in the case of Niho exponents. We also state a conjecture for when the Weil spectrum contains at least five elements and prove it for a certain class of Weil sums.