Orthomorphism Polynomials of degree $7$ over finite fields

TL;DR

This paper leverages a 2019 classification of degree 7 permutation polynomials over finite fields to identify all degree 7 orthomorphism polynomials over specific finite fields and proves their non-existence in some cases.

Contribution

It provides a complete list of degree 7 orthomorphism polynomials over select finite fields and establishes non-existence results for certain fields.

Findings

Complete list of degree 7 orthomorphism polynomials over specified fields

Non-existence of these polynomials in some finite fields

Application of previous classification to new polynomial types

Abstract

In 2019, Xiang Fan \cite{xfan} classified all permutation polynomials of degree $7$ over finite fields of odd characteristics. In this paper, we use this classification to determine the complete list of degree $7$ orthomorphism polynomials over finite fields of order $q\in\{11,~13,~17,~19,~25,~49\}.$ In addition, the non-existence of these polynomials is established for certain fields.