The Multiple Equal-Difference Structure of Cyclotomic Cosets

2025/01/07 by Li Zhu, Juncheng Zhou, Zhu, Li +4 · 1 citation
Engineering · Computer Science · #graph theory and CDMA systems #Advanced Graph Theory Research #Graph Labeling and Dimension Problems

paper · pdf · doi:10.48550/arxiv.2501.03516

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 Xⁿ-1 in binomial form over finite extension fields of \mathbbF_q. We give an explicit characterization of the multiple equal-difference representations of any q-cyclotomic coset modulo n, through which a criterion for Xⁿ-1 factoring into irreducible binomials is obtained. In addition, we present an algorithm to simplify the computation of the leaders of cyclotomic cosets.

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• The Minimal Binomial Multiples of Polynomials over Finite Fields

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