Rationality of Four-Valued Families of Weil Sums of Binomials
Daniel J. Katz, Allison E. Wong
TL;DR
The paper addresses when the Weil spectrum of binomial Weil sums over finite fields is rational, focusing on the 4-valued case. It combines algebraic number theory, bounds, algebraic-geometry over finite fields, group-algebra techniques, and cyclotomic actions to constrain the possible value distributions, culminating in a precise rigidity result: a 4-valued spectrum is rational unless the field is $\mathbb{F}_5$ and $s \equiv 3 \pmod{4}$, in which case the spectrum is explicitly $\{(5 \pm \sqrt{5})/2, \pm \sqrt{5}\}$. The work shows that all four values are algebraic integers with strong $p$-adic and Galois constraints, forcing integrality in all but the exceptional case. This advances the understanding of Walsh spectra and their arithmetic structure, with implications for related code and sequence design problems.
Abstract
We investigate the rationality of Weil sums of binomials of the form $W^{K,s}_u=\sum_{x \in K} ψ(x^s - u x)$, where $K$ is a finite field whose canonical additive character is $ψ$, and where $u$ is an element of $K^{\times}$ and $s$ is a positive integer relatively prime to $|K^\times|$, so that $x \mapsto x^s$ is a permutation of $K$. The Weil spectrum for $K$ and $s$, which is the family of values $W^{K,s}_u$ as $u$ runs through $K^\times$, is of interest in arithmetic geometry and in several information-theoretic applications. The Weil spectrum always contains at least three distinct values if $s$ is nondegenerate (i.e., if $s$ is not a power of $p$ modulo $|K^\times|$, where $p$ is the characteristic of $K$). It is already known that if the Weil spectrum contains precisely three distinct values, then they must all be rational integers. We show that if the Weil spectrum contains precisely four distinct values, then they must all be rational integers, with the sole exception of the case where $|K|=5$ and $s \equiv 3 \pmod{4}$.