Uniqueness of the inversive plane of order sixty-four

Penttila, Tim

doi:10.1007/s10623-022-01014-6

Cited by 5 publications

(3 citation statements)

References 36 publications

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“…Since every oval lies in a unique hyperoval, all the ovals in PG(2, 64), up to equivalence, can be obtained by removing one "representative" from each point-orbit of the stabilizer in PΓL(3, 64) of each hyperoval, considered as a permutation group acting on it. This operation produces 19 isomorphism classes of ovals in PG(2, 64) (obtained by Siciliano on behalf of Penttila for a paper on the uniqueness of the inversive plane of order 64 [48]). By Theorem 2.5, the magic action produces all o-polynomials over GF(64), split in 19 classes.…”

Section: Computational Resultsmentioning

confidence: 99%

Classification of flocks of the quadratic cone in PG(3,64)

Giusy¹,

Penttila²,

Siciliano³

2022

Preprint

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Flocks are an important topic in the field of finite geometry, with many relations with other objects of interest. This paper is a contribution to the * The research was supported by the Italian National Group for Algebraic and Geometric Structures and their Applications (GNSAGA-INdAM). difficult problem of classifying flocks up to projective equivalence. We complete the classification of flocks of the quadratic cone in PG(3, q) for q ≤ 71, by showing by computer that there are exactly three flocks of the quadratic cone in PG(3, 64), up to equivalence. The three flocks had previously been discovered, and they are the linear flock, the Subiaco flock and the Adelaide flock. The classification proceeds via the connection between flocks and herds of ovals in PG(2, q), q even, and uses the prior classification of hyperovals in PG(2, 64).

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Section: Computational Resultsmentioning

confidence: 99%

Classification of flocks of the quadratic cone in PG(3,64)

Giusy¹,

Penttila²,

Siciliano³

2022

Preprint

Self Cite

View full text Add to dashboard Cite

show abstract

“…Since every oval lies in a unique hyperoval, all the ovals in PG(2, 64), up to equivalence, can be obtained by removing one representative from each point-orbit of the stabilizer in PΓL(3, 64) of each hyperoval, considered as a permutation group acting on it. This operation produces exactly 19 distinct ovals in PG(2, 64) [20].…”

Section: The Classificationmentioning

confidence: 99%

“…The recent classification of hyperovals of PG (2,64) [28] has provided an opportunity to extend classification results for related structures to order 64. The first of these was inversive planes [20], which correspond to fans of ovals. The second of these was flocks (and the related elation generalized quadrangles) [11], which correspond to herds of ovals.…”

Section: Introductionmentioning

confidence: 99%