🤖 AI Summary
This paper addresses the structural characterization of cyclotomic cosets over finite fields and the binomial irreducible factorization of $X^n - 1$. We introduce the notion of *arithmetic cyclotomic cosets* and establish, for the first time, a *multiple arithmetic progression representation theory* for cyclotomic cosets: every $q$-cyclotomic coset modulo $n$ admits a unique decomposition into disjoint arithmetic progressions, and this decomposition is in one-to-one correspondence with the complete factorization of $X^n - 1$ into irreducible binomials over $mathbb{F}_q[X]$. Building upon this theory, we provide an explicit construction algorithm, necessary and sufficient criteria for such factorizations—including a generalization to extension fields $mathbb{F}_{q^m}$—and a novel efficient algorithm for computing minimal representatives of cyclotomic cosets. Our results unify combinatorial coset analysis with algebraic polynomial factorization, yielding new tools for cyclic code design and finite-field polynomial decomposition.
📝 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 represent an algorithm to simplify the computation of the leaders of cyclotomic cosets.