On Weil Sums, Conjectures of Helleseth and Niho Exponents

28 Jun 2020  ·  Nguyen Liem ·

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 $W_{F,s}(a)=\displaystyle\sum_{x \in F^\times} \mu(x^s-ax)$... 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 $1971$, Helleseth formulated two conjectures about the presence of zero value in the Weil spectrum and the size of the spectrum, both of which still remain unsolved. In this paper, we discuss some progress made towards these two conjectures in the case of Niho exponents. We also give a conjecture considering the Weil spectrum of at least five elements, and discuss some partial results. read more

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Number Theory