Proofs of two conjectures on ternary weakly regular bent functions

TL;DR

This paper proves that specific ternary monomial functions and Coulter-Matthews bent functions are weakly regular, confirming conjectures and advancing understanding of ternary bent functions using number-theoretic tools.

Contribution

It establishes the weak regularity of certain ternary monomial and Coulter-Matthews bent functions, settling prior conjectures in the field.

Findings

Certain ternary monomial functions are weakly regular bent.

Coulter-Matthews bent functions are weakly regular.

The results confirm conjectures by Helleseth and Kholosha.

Abstract

We study ternary monomial functions of the form $f(x)=\Tr_n(ax^d)$, where $x\in \Ff_{3^n}$ and $\Tr_n: \Ff_{3^n}\to \Ff_3$ is the absolute trace function. Using a lemma of Hou \cite{hou}, Stickelberger's theorem on Gauss sums, and certain ternary weight inequalities, we show that certain ternary monomial functions arising from \cite{hk1} are weakly regular bent, settling a conjecture of Helleseth and Kholosha \cite{hk1}. We also prove that the Coulter-Matthews bent functions are weakly regular.