The Minimal Binomial Multiples of Polynomials over Finite Fields
Li Zhu, Hongfeng Wu
TL;DR
The paper introduces the minimal binomial multiple of a nonconstant polynomial f(X) over 𝔽_q, defined as the monic binomial X^n−λ of smallest degree with f(X) | X^n−λ, and formalizes ord^b(f)=n. It develops a radical/defining-set framework to generalize the classical order, yielding explicit formulas for the minimal binomial multiple in terms of the defining set of rad(f) and a criterion for freeness (ord^b(f)=ord(f)). It provides complete factorization tools for binomials via q-cyclotomic cosets, giving explicit irreducible factorizations of X^N−λ and a systematic method to compute minimal binomial multiples in both squarefree and general cases. As an application, it characterizes λ-constacyclic codes of length N with minimal distance 2 by linking code divisors to the minimal binomial multiple, enabling exact classification in a broad range of scenarios.
Abstract
Let $f(X)$ be a nonconstant polynomial over $\mathbb{F}_{q}$, with a nonzero constant term. The order of $f(X)$ is a classical notion in the theory of polynomials over finite fields, and recently the definition of freeness of binomials of $f(X)$ was given in \cite{Martínez}. Generalizing these two notions, we introduce the definition of the minimal binomial multiple of $f(X)$ in this paper, which is the monic binomial with the lowest degree among the binomials over $\mathbb{F}_{q}$ divided by $f(X)$. Based on the equivalent characterization of binomials via the defining sets of their radicals, we prove that a series of properties of the classical order can be naturally generalized to this case. In particular, the minimal binomial multiple of $f(X)$ is presented explicitly in terms of the defining set of the radical of $f(X)$. And a criterion for $f(X)$ being free of binomials is given. As an application, for any positive integer $N$ and nonzero element $λ$ in $\mathbb{F}_{q}$, the $λ$-constacyclic codes of length $N$ with minimal distance $2$ are determined.