The Minimal Binomial Multiples of Polynomials over Finite Fields

TL;DR

This paper introduces the concept of minimal binomial multiples of polynomials over finite fields, generalizes classical notions like order and freeness, and applies these ideas to classify certain constacyclic codes.

Contribution

It defines the minimal binomial multiple, provides explicit formulas, and establishes criteria for freeness, extending classical polynomial properties over finite fields.

Findings

Explicit formula for minimal binomial multiple in terms of radical defining set

Criterion for polynomial freeness regarding binomials

Classification of minimal distance 2 constacyclic codes

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{Mart\'{i}nez}. 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 $\lambda$ in $\mathbb{F}_{q}$, the $\lambda$-constacyclic codes of length $N$ with minimal distance $2$ are determined.