Existence results for cyclotomic orthomorphisms

TL;DR

This paper characterizes the existence of cyclotomic orthomorphisms over finite fields, solves open problems about their minimal index, and constructs orthogonal sets of such orthomorphisms for large fields.

Contribution

It provides a complete characterization of when cyclotomic orthomorphisms of a given least index exist over finite fields and constructs large orthogonal families for various indices.

Findings

Existence of cyclotomic orthomorphisms characterized for all pairs (q,k).

Large orthogonal families of orthomorphisms constructed for sufficiently large fields.

Exact conditions established for small index pairs and counts of linear orthomorphisms.

Abstract

An {\em orthomorphism} over a finite field $\mathbb{F}$ is a permutation $\theta:\mathbb{F}\mapsto\mathbb{F}$ such that the map $x\mapsto\theta(x)-x$ is also a permutation of $\mathbb{F}$. The orthomorphism $\theta$ is {\em cyclotomic of index $k$} if $\theta(0)=0$ and $\theta(x)/x$ is constant on the cosets of a subgroup of index $k$ in the multiplicative group $\mathbb{F}^*$. We say that $\theta$ has {\em least index} $k$ if it is cyclotomic of index $k$ and not of any smaller index. We answer an open problem due to Evans by establishing for which pairs $(q,k)$ there exists an orthomorphism over $\mathbb{F}_q$ that is cyclotomic of least index $k$. Two orthomorphisms over $\mathbb{F}_q$ are orthogonal if their difference is a permutation of $\mathbb{F}_q$. For any list $[b_1,\dots,b_n]$ of indices we show that if $q$ is large enough then $\mathbb{F}_q$ has pairwise orthogonal orthomorphisms of least indices $b_1,\dots,b_n$. This provides a partial answer to another open problem due to Evans. For some pairs of small indices we establish exactly which fields have orthogonal orthomorphisms of those indices. We also find the number of linear orthomorphisms that are orthogonal to certain cyclotomic orthomorphisms of higher index.