On The Number Of Quadratic Orthomorphisms That Produce Maximally Nonassociative Quasigroups

Aleš Drápal; Ian M. Wanless

doi:10.1017/S1446788722000386

On The Number Of Quadratic Orthomorphisms That Produce Maximally Nonassociative Quasigroups

Aleš Drápal, Ian M. Wanless

School of Mathematics

Research output: Contribution to journal › Article › Research › peer-review

2 Citations (Scopus)

Abstract

Let q be an odd prime power and suppose that are such that and are nonzero squares. Let be the quasigroup in which the operation is defined by if is a square, and if is a nonsquare. This quasigroup is called maximally nonassociative if it satisfies. Denote by the number of for which is maximally nonassociative. We show that there exist constants and such that if, then, and if, then.

Original language	English
Pages (from-to)	311-336
Number of pages	26
Journal	Journal of the Australian Mathematical Society
Volume	115
Issue number	3
DOIs	https://doi.org/10.1017/S1446788722000386
Publication status	Published - 20 Dec 2023

Keywords

finite field
maximally nonassociative
quadratic orthomorphism
quasigroup

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10.1017/S1446788722000386Licence: CC BY

Cite this

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abstract = "Let q be an odd prime power and suppose that are such that and are nonzero squares. Let be the quasigroup in which the operation is defined by if is a square, and if is a nonsquare. This quasigroup is called maximally nonassociative if it satisfies. Denote by the number of for which is maximally nonassociative. We show that there exist constants and such that if, then, and if, then.",

keywords = "finite field, maximally nonassociative, quadratic orthomorphism, quasigroup",

author = "Ale{\v s} Dr{\'a}pal and Wanless, \{Ian M.\}",

note = "Funding Information: This work was supported in part by Australian Research Council grant DP150100506. Publisher Copyright: {\textcopyright} 2023 The Author(s). Published by Cambridge University Press on behalf of Australian Mathematical Publishing Association Inc.",

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Y1 - 2023/12/20

N2 - Let q be an odd prime power and suppose that are such that and are nonzero squares. Let be the quasigroup in which the operation is defined by if is a square, and if is a nonsquare. This quasigroup is called maximally nonassociative if it satisfies. Denote by the number of for which is maximally nonassociative. We show that there exist constants and such that if, then, and if, then.

AB - Let q be an odd prime power and suppose that are such that and are nonzero squares. Let be the quasigroup in which the operation is defined by if is a square, and if is a nonsquare. This quasigroup is called maximally nonassociative if it satisfies. Denote by the number of for which is maximally nonassociative. We show that there exist constants and such that if, then, and if, then.

KW - finite field

KW - maximally nonassociative

KW - quadratic orthomorphism

KW - quasigroup

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On The Number Of Quadratic Orthomorphisms That Produce Maximally Nonassociative Quasigroups

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Matchings in Combinatorial Structures

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On The Number Of Quadratic Orthomorphisms That Produce Maximally Nonassociative Quasigroups

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Keywords

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Matchings in Combinatorial Structures

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