Degree of Orthomorphism Polynomials over Finite Fields

TL;DR

This paper investigates the maximum degree of orthomorphism polynomials over finite fields, proving the upper bound is achieved in almost all cases and constructing related combinatorial objects.

Contribution

It establishes that the maximum degree of orthomorphisms is attained for all prime powers except a few small cases, using novel constructions involving minimal differences.

Findings

Maximum degree of orthomorphisms is q-3 for most finite fields.

Constructs two orthomorphisms differing on only three elements.

Applications to constructing 3-homogeneous Latin bitrades.

Abstract

An orthomorphism over a finite field $\mathbb{F}_q$ is a permutation $\theta:\mathbb{F}_q\mapsto\mathbb{F}_q$ such that the map $x\mapsto\theta(x)-x$ is also a permutation of $\mathbb{F}_q$. The degree of an orthomorphism of $\mathbb{F}_q$, that is, the degree of the associated reduced permutation polynomial, is known to be at most $q-3$. We show that this upper bound is achieved for all prime powers $q\notin\{2, 3, 5, 8\}$. We do this by finding two orthomorphisms in each field that differ on only three elements of their domain. Such orthomorphisms can be used to construct $3$-homogeneous Latin bitrades.