Beyond uniform cyclotomy

TL;DR

This work tackles the difficulty of obtaining explicit cyclotomic numbers beyond small orders by extending uniform cyclotomy into a general, combinatorial framework for all $e\mid (q^n-1)/(q-1)$. It leverages the vector-space view of ${\rm GF}(q^n)$ over ${\rm GF}(q)$, Singer difference sets, and finite geometry to derive direct evaluations that avoid character sums and field computations. The authors provide explicit descriptions for the main case $e=(q^n-1)/(q-1)$, show how $S_k$ can be computed via hyperplane intersections (and subfield reductions when appropriate), and extend to orders $\epsilon\mid e$ through a partitioning approach. The techniques enable practical computation of many cyclotomic numbers previously inaccessible, with concrete implications for difference structures and related combinatorial constructions.

Abstract

Cyclotomy, the study of cyclotomic classes and cyclotomic numbers, is an area of number theory first studied by Gauss. It has natural applications in discrete mathematics and information theory. Despite this long history, there are significant limitations to what is known explicitly about cyclotomic numbers, which limits the use of cyclotomy in applications. The main explicit tool available is that of uniform cyclotomy, introduced by Baumert, Mills and Ward in 1982. In this paper, we present an extension of uniform cyclotomy which gives a direct method for evaluating all cyclotomic numbers over $GF(q^n)$ of order dividing $(q^n-1)/(q-1)$, for any prime power $q$ and $n \geq 2$, which does not use character theory nor direct calculation in the field. This allows the straightforward evaluation of many cyclotomic numbers for which other methods are unknown or impractical. Our methods exploit connections between cyclotomy, Singer difference sets and finite geometry.

Theorems & Definitions (46)

• Definition 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• proof
• Definition 2.1
• Definition 2.2
• Lemma 2.3
• Definition 2.4
• Theorem 2.5: Singer