Maximal nonassociativity via nearfields

Drápal, Aleš; Lisonĕk, P

doi:10.1016/j.ffa.2019.101610

Cited by 6 publications

(37 citation statements)

References 6 publications

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“…In this paper only the case of 9 elements is discussed. That case exhibits a specific behavior, and that makes the intersection with quite thin.…”

Section: The Nearfield Connectionmentioning

confidence: 95%

“…When a quadratic nearfield

N

of an order

q^{2} > 9

is considered, then it is not true that every

*_{c}, c \in N \ F_{q}

, yields an extremely nonassociative quasigroup. Nevertheless there always exists an element

c \in N \ F_{q}

such that

(N {, *}_{c})

is extremely nonassociative . The proof uses a probabilistic argument.…”

Section: The Nearfield Connectionmentioning

confidence: 97%

“…However, in the general setting there seems to be no clue how to choose

c

for which

(N {, *}_{c})

is extremely nonassociative. Computer experiments suggest that the number of such

c

converges to

λ q^{2}

where

0.288 < λ < 0.29

. There is some hope that the proof of Theorem can be generalized in a way that would yield results of in a deterministic manner.…”

Section: The Nearfield Connectionmentioning

confidence: 99%

“…Computer experiments suggest that the number of such

c

converges to

λ q^{2}

where

0.288 < λ < 0.29

. There is some hope that the proof of Theorem can be generalized in a way that would yield results of in a deterministic manner.…”

Section: The Nearfield Connectionmentioning

confidence: 99%

“…In particular, a quadratic nearfield of order

q^{2}

gives rise to such a quasigroup for each odd prime power

q

. This is the content of . In this paper only the case of 9 elements is discussed.…”

Section: The Nearfield Connectionmentioning

confidence: 99%

See 4 more Smart Citations

Extreme nonassociativity in order nine and beyond

Drápal

Valent

2019

J of Combinatorial Designs

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“…In this paper only the case of 9 elements is discussed. That case exhibits a specific behavior, and that makes the intersection with quite thin.…”

Section: The Nearfield Connectionmentioning

confidence: 95%

“…When a quadratic nearfield

N

of an order

q^{2} > 9

is considered, then it is not true that every

*_{c}, c \in N \ F_{q}

, yields an extremely nonassociative quasigroup. Nevertheless there always exists an element

c \in N \ F_{q}

such that

(N {, *}_{c})

is extremely nonassociative . The proof uses a probabilistic argument.…”

Section: The Nearfield Connectionmentioning

confidence: 97%

“…However, in the general setting there seems to be no clue how to choose

c

for which

(N {, *}_{c})

is extremely nonassociative. Computer experiments suggest that the number of such

c

converges to

λ q^{2}

where

0.288 < λ < 0.29

. There is some hope that the proof of Theorem can be generalized in a way that would yield results of in a deterministic manner.…”

Section: The Nearfield Connectionmentioning

confidence: 99%

“…Computer experiments suggest that the number of such

c

converges to

λ q^{2}

where

0.288 < λ < 0.29

. There is some hope that the proof of Theorem can be generalized in a way that would yield results of in a deterministic manner.…”

Section: The Nearfield Connectionmentioning

confidence: 99%

“…In particular, a quadratic nearfield of order

q^{2}

gives rise to such a quasigroup for each odd prime power

q

. This is the content of . In this paper only the case of 9 elements is discussed.…”

Section: The Nearfield Connectionmentioning

confidence: 99%

See 3 more Smart Citations

Extreme nonassociativity in order nine and beyond

Drápal

Valent

2019

J of Combinatorial Designs

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Maximal nonassociativity via fields

Lisonĕk

2020

Des. Codes Cryptogr.

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On the Number of Quadratic Orthomorphisms That Produce Maximally Nonassociative Quasigroups

Drápal¹,

Wanless

2023

J. Aust. Math. Soc.

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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 $u*v=u+b(v{-}u)$ if $v-u$ 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 $\alpha \approx 0.029\,08$ and $\beta \approx 0.012\,59$ 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 $ .

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Maximal nonassociativity via nearfields

Cited by 6 publications

References 6 publications

Extreme nonassociativity in order nine and beyond

Extreme nonassociativity in order nine and beyond

Maximal nonassociativity via fields

On the Number of Quadratic Orthomorphisms That Produce Maximally Nonassociative Quasigroups

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