Computational Results on the Existence of Primitive Complete Normal Basis Generators

TL;DR

This paper provides computational evidence supporting a longstanding conjecture that in Galois field extensions, primitive elements that are completely normal over the base field always exist.

Contribution

The authors offer computational results that strongly support Morgan and Mullen's conjecture regarding primitive completely normal basis generators in Galois fields.

Findings

Computational results support the conjecture for various Galois extensions.

Evidence suggests the universal existence of primitive completely normal elements.

Supports the conjecture's validity across multiple cases.

Abstract

We present computational results which strongly support a conjecture of Morgan and Mullen (1996), which states that for every extension $E/F$ of Galois fields there exists a primitive element of $E$ which is completely normal over $F$.