On partial geometries arising from maximal arcs

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dc.authorscopusid57201668033
dc.authorscopusid7003357967
dc.contributor.authorGezek, Mustafa
dc.contributor.authorTonchev, Vladimir D.
dc.date.accessioned2022-05-11T14:04:43Z
dc.date.available2022-05-11T14:04:43Z
dc.date.issued2022
dc.departmentFakülteler, Fen Edebiyat Fakültesi, Matematik Bölümü
dc.description.abstractThe subject of this paper are partial geometries pg(s, t, ?) with parameters s=d(d?-1),t=d?(d-1),?=(d-1)(d?-1), d, d?? 2. In all known examples, q= dd? is a power of 2 and the partial geometry arises from a maximal arc of degree d or d? in a projective plane of order q via a known construction due to Thas [28] and Wallis [34], with a single known exception of a partial geometry pg(4, 6, 3) found by Mathon [22] that is not associated with a maximal arc in the projective plane of order 8. A parallel class of lines is a set of pairwise disjoint lines that covers the point set. Two parallel classes are called orthogonal if they share exactly one line. An upper bound on the maximum number of pairwise orthogonal parallel classes in a partial geometry G with parameters pg(d(d?- 1) , d?(d- 1) , (d- 1) (d?- 1)) is proved, and it is shown that a necessary and sufficient condition for G to arise from a maximal arc of degree d or d? in a projective plane of order q= dd? is that both G and its dual geometry contain sets of pairwise orthogonal parallel classes that meet the upper bound. An alternative construction of Mathon’s partial geometry is presented, and the new necessary condition is used to demonstrate why this partial geometry is not associated with any maximal arc in the projective plane of order 8. The partial geometries associated with all known maximal arcs in projective planes of order 16 are classified up to isomorphism, and their parallel classes of lines and the 2-rank of their incidence matrices are computed. Based on these results, some open problems and conjectures are formulated. © 2021, Springer Science+Business Media, LLC, part of Springer Nature.
dc.description.sponsorshipThe authors thank the anonymous reviewers for reading carefully the manuscript and for their helpful suggestions.
dc.identifier.doi10.1007/s10801-020-00995-8
dc.identifier.endpage139
dc.identifier.issn0925-9899
dc.identifier.issue1en_US
dc.identifier.scopus2-s2.0-85099090009
dc.identifier.scopusqualityQ1
dc.identifier.startpage117
dc.identifier.urihttps://doi.org/10.1007/s10801-020-00995-8
dc.identifier.urihttps://hdl.handle.net/20.500.11776/4737
dc.identifier.volume55
dc.identifier.wosWOS:000605899600002
dc.identifier.wosqualityQ3
dc.indekslendigikaynakWeb of Science
dc.indekslendigikaynakScopus
dc.institutionauthorGezek, Mustafa
dc.language.isoen
dc.publisherSpringer
dc.relation.ispartofJournal of Algebraic Combinatorics
dc.relation.publicationcategoryMakale - Uluslararası Hakemli Dergi - Kurum Öğretim Elemanıen_US
dc.rightsinfo:eu-repo/semantics/openAccess
dc.subjectMaximal arc
dc.subjectPartial geometry
dc.subjectProjective plane
dc.subjectStrongly regular graph
dc.titleOn partial geometries arising from maximal arcs
dc.typeArticle

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