All bi-unitary perfect polynomials over $\F_2$ only divisible by $x$, $x+1$ and by Mersenne primes

TL;DR

This paper classifies all bi-unitary perfect polynomials over GF(2) that are composed solely of Mersenne polynomials and only divisible by x, x+1, and Mersenne primes, providing a complete characterization.

Contribution

It provides a complete classification of non-splitting bi-unitary perfect polynomials over GF(2) with specific divisibility conditions, focusing on Mersenne polynomial factors.

Findings

All such bi-unitary perfect polynomials are explicitly characterized.

The polynomials are only divisible by x, x+1, and Mersenne primes.

The classification completes the understanding of these special polynomials over GF(2).

Abstract

We give all non splitting bi-unitary perfect polynomials over the prime field of two elements, which have only Mersenne polynomials as odd irreducible divisors.