Cyclotomic Numbers of Order $q-1$ over $\mathbb{F}_{q^r}$

TL;DR

This paper investigates bounds on cyclotomic numbers of order q-1 over finite fields, providing general and sharper bounds for specific cases, notably for prime r.

Contribution

It establishes new upper bounds for cyclotomic numbers over finite fields, including sharper bounds for prime r, especially r=2 and r=3.

Findings

Bound (a,b)_{q-1} ≤ ⌈k/2⌉ for all cases except q=2, r≥3.

Sharper bounds are derived for prime r, notably r=2 and r=3.

The bounds improve understanding of cyclotomic number distributions over finite fields.

Abstract

Let $q=p^n$, $r\in \mathbb{Z}_{\ge 2}$, $e=q-1$, and $k=\frac{q^r-1}{e}$. In this paper, we study the cyclotomic numbers $(a,b)_{q-1}$ over $\mathbb{F}_{q^r}$. We prove that $(a,b)_{q-1}\le \left\lceil \frac{k}{2}\right\rceil$ for all $0\le a,b\le q-2$ except when $q=2$ and $r\ge 3$. We also give sharper bounds for prime values of $r$, especially for $r=2$ and $r=3$.