Orthomorphism Polynomials of degree $7$ over finite fields

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.

14 pages