The Multiple Equal-Difference Structure of Cyclotomic Cosets

TL;DR

This paper introduces the concept of equal-difference cyclotomic cosets, explores their decompositions, and establishes a link between these decompositions and the factorization of polynomials over finite fields, providing new tools for algebraic analysis.

Contribution

It defines equal-difference cyclotomic cosets, characterizes their multiple decompositions, and connects these to polynomial factorizations, offering explicit criteria and algorithms.

Findings

Decomposition of cyclotomic cosets into equal-difference subsets.

Correspondence between coset decompositions and polynomial factorizations.

Algorithm for computing cyclotomic coset leaders.

Abstract

In this paper we introduce the definition of equal-difference cyclotomic coset, and prove that in general any cyclotomic coset can be decomposed into a disjoint union of equal-difference subsets. Among the equal-difference decompositions of a cyclotomic coset, an important class consists of those in the form of cyclotomic decompositions, called the multiple equal-difference representations of the coset. There is an equivalent correspondence between the multiple equal-difference representations of $q$-cyclotomic cosets modulo $n$ and the irreducible factorizations of $X^{n}-1$ in binomial form over finite extension fields of $\mathbb{F}_{q}$. We give an explicit characterization of the multiple equal-difference representations of any $q$-cyclotomic coset modulo $n$, through which a criterion for $X^{n}-1$ factoring into irreducible binomials is obtained. In addition, we present an algorithm to simplify the computation of the leaders of cyclotomic cosets.