Admissible family for binary perfect polynomials

TL;DR

This paper establishes necessary conditions for the prime divisors of perfect polynomials over _2, providing a new characterization of known perfect polynomials and a pathway to discover new ones if they exist.

Contribution

It introduces admissible conditions for prime divisors of perfect polynomials over _2 and offers a new characterization of all known perfect polynomials.

Findings

Characterization of all known perfect polynomials over _2

Necessary conditions for prime divisors of perfect polynomials

Potential method for discovering new perfect polynomials

Abstract

The paper is about an arithmetic problem in $\F_2[x]$. We give \emph{admissible} (necessary) conditions satisfied by a set of odd prime divisors of perfect polynomials over $\F_2$. This allows us to prove a new characterization of \emph{all} known perfect polynomials, and to open a way of finding more of them (if they exist).