On the number of quadratic orthomorphisms that produce maximally nonassociative quasigroups

TL;DR

This paper investigates the count of quadratic orthomorphisms over finite fields that generate maximally nonassociative quasigroups, establishing asymptotic densities depending on the congruence class of the prime power.

Contribution

It provides the first asymptotic formulas for the number of such orthomorphisms, revealing distinct limiting densities based on the field's congruence class.

Findings

For $q ot rsim 1 mod 4$, the density approaches approximately 0.02908.

For $q ot rsim 3 mod 4$, the density approaches approximately 0.01259.

The results quantify the distribution of maximally nonassociative quadratic orthomorphisms over finite fields.

Abstract

Let $q$ be an odd prime power and suppose that $a,b\in\mathbb{F}_q$ are such that $ab$ and $(1{-}a)(1{-}b)$ are nonzero squares. Let $Q_{a,b} = (\mathbb{F}_q,*)$ be the quasigroup in which the operation is defined by $u*v=u+a(v{-}u)$ if $v-u$ is a square, and $u*v=u+b(v{-}u)$ is $v-u$ is a nonsquare. This quasigroup is called maximally nonassociative if it satisfies $x*(y*z) = (x*y)*z$ $\Leftrightarrow$ $x=y=z$. Denote by $\sigma(q)$ the number of $(a,b)$ for which $Q_{a,b}$ is maximally nonassociative. We show that there exist constants $\alpha \approx 0.02908$ and $\beta \approx 0.01259$ such that if $q\equiv 1 \bmod 4$, then $\lim \sigma(q)/q^2 = \alpha$, and if $q \equiv 3 \bmod 4$, then $\lim \sigma(q)/q^2 = \beta$.