Non-splitting bi-unitary perfect polynomials over $\mathbb{F}_4$ with less than five prime factors

TL;DR

This paper classifies all non-splitting bi-unitary perfect polynomials over the finite field  with up to four prime factors, revealing an infinite family of such polynomials.

Contribution

It provides a complete classification of non-splitting bi-unitary perfect polynomials over  with limited prime factors, including the discovery of infinitely many such polynomials.

Findings

All such polynomials are identified and classified.

An infinite number of these polynomials exist.

The classification covers polynomials with up to four prime factors.

Abstract

We identify all non-splitting bi-unitary perfect polynomials over the field $\mathbb{F}_4$, which admit at most four irreducible divisors. There is an infinite number of such divisors.