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- W4287775823 abstract "Let $q$ be an odd prime power and suppose that $a,binmathbb{F}_q$ 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 $qequiv 1 bmod 4$, then $lim sigma(q)/q^2 = alpha$, and if $q equiv 3 bmod 4$, then $lim sigma(q)/q^2 = beta$." @default.
- W4287775823 created "2022-07-26" @default.
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- W4287775823 date "2020-05-24" @default.
- W4287775823 modified "2023-09-28" @default.
- W4287775823 title "On the number of quadratic orthomorphisms that produce maximally nonassociative quasigroups" @default.
- W4287775823 doi "https://doi.org/10.48550/arxiv.2005.11674" @default.
- W4287775823 hasPublicationYear "2020" @default.
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