On the number of quadratic orthomorphisms that produce maximally nonassociative quasigroups

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 $\lim \sigma(q)/q^2 = \beta$.