Permutation Polynomials of Degree 6 or 7 over Finite Fields of Characteristic 2

TL;DR

This paper completes the classification of permutation polynomials of degrees 6 and 7 over finite fields of characteristic 2, extending previous work limited to odd characteristic fields.

Contribution

It provides a complete classification of degree 6 and 7 permutation polynomials over finite fields of characteristic 2, filling a gap in existing literature.

Findings

Complete classification of degree 6 permutation polynomials over characteristic 2 fields.

Classification of all degree 7 permutation polynomials over characteristic 2 fields.

Extends previous classifications limited to odd characteristic fields.

Abstract

In \cite{D1}, Dickson listed all permutation polynomials up to degree 5 over an arbitrary finite field, and all permutation polynomials of degree 6 over finite fields of odd characteristic. The classification of degree 6 permutation polynomials over finite fields of characteristic 2 was left incomplete. In this paper we complete the classification of permutation polynomials of degree 6 over finite fields of characteristic 2. In addition, all permutation polynomials of degree 7 over finite fields of characteristic 2 are classified.