On partial geometries arising from maximal arcs
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Dosyalar
Tarih
2022
Yazarlar
Dergi Başlığı
Dergi ISSN
Cilt Başlığı
Yayıncı
Springer
Erişim Hakkı
info:eu-repo/semantics/openAccess
Özet
The 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.
Açıklama
Anahtar Kelimeler
Maximal arc, Partial geometry, Projective plane, Strongly regular graph
Kaynak
Journal of Algebraic Combinatorics
WoS Q Değeri
Q3
Scopus Q Değeri
Q1
Cilt
55
Sayı
1
