On bi-unitary perfect polynomials over $F_2$

TL;DR

This paper classifies all bi-unitary perfect polynomials over the finite field with two elements, focusing on those divisible by Mersenne irreducibles and with limited irreducible factors, completing previous classifications.

Contribution

It provides a complete classification of bi-unitary perfect polynomials over F_2, including those divisible by Mersenne irreducibles and with up to four irreducible factors.

Findings

All bi-unitary non-splitting even perfect polynomials over F_2 identified.

All bi-unitary perfect polynomials with up to four irreducible factors classified.

Extended previous classifications by J.T. B. Beard Jr.

Abstract

We give all bi-unitary non splitting even perfect polynomials over the prime field of two elements, which are divisible by Mersenne irreducible polynomials raised to special exponents. We also identify all bi-unitary perfect polynomials over the same field, with at most four irreducible factors. We then complete, in this manner, a list given by J.T. B. Beard Jr.