Sets of type \((m,n)\) in the affine and projective planes of order nine

DOI10.1007/BF01388477zbMath0841.05012OpenAlexW2059895764WikidataQ59411507 ScholiaQ59411507MaRDI QIDQ1904423

Tim Penttila, Gordon F. Royle

Publication date: 20 December 1995

Published in: Designs, Codes and Cryptography (Search for Journal in Brave)

Full work available at URL: https://doi.org/10.1007/bf01388477

zbMATH Keywords

projective planeslineaffine planespoints of type \((m, n)\)

Mathematics Subject Classification ID

Combinatorial aspects of finite geometries (05B25)

Related Items (38)

Quasimultiples of geometric designs ⋮ Sets of type-\((1,n)\) in symmetric designs for \(\lambda \geq 3\) ⋮ The non-existence of 3-dimensional locally projective spaces of orders (2,9) ⋮ Triangle-free induced subgraphs of the unitary polarity graph ⋮ Some new two-weight ternary and quinary codes of lengths six and twelve ⋮ Maximal arcs in Desarguesian planes of odd order do not exist ⋮ Embedding of classical polar unitals in \(\mathrm{PG}(2,q^{2})\) ⋮ On standard two-intersection sets in \(\mathrm{PG}(r,q)\) ⋮ Unitals in the Desarguesian projective plane of order 16 ⋮ Unitals in projective planes of order 16 ⋮ Unitals in projective planes of order 25 ⋮ Unital designs with blocking sets ⋮ Cameron-Liebler line classes ⋮ On sets with few intersection numbers in finite projective and affine spaces ⋮ Unital designs in planes of order 16 ⋮ Baer subgeometry partitions ⋮ Minimal multiple blocking sets ⋮ Unitals with many Baer secants through a fixed point ⋮ A combinatorial characterization of parabolic quadrics ⋮ A characterization of the set of internal points of a conic in \(\mathrm{PG}(2,q), q\) odd ⋮ Unitals of PG(2,q²) Containing Conics ⋮ Ternary dual codes of the planes of order nine ⋮ Note on a class of subsets of \(\mathrm{AG}(3, q)\) with intersection numbers \(1\), \(q\) and \(n\) with respect to the planes ⋮ Strongly regular \((\alpha,\beta)\)-geometries ⋮ Constructing two-weight codes with prescribed groups of automorphisms ⋮ Preface ⋮ An Enumeration of Certain Projective Ternary Two-Weight Codes ⋮ On multiple blocking sets and blocking semiovals in finite non-Desarguesian planes ⋮ The Kramer-Mesner method with tactical decompositions: some new unitals on 65 points ⋮ Embedding of orthogonal Buekenhout-Metz unitals in the Desarguesian plane of order q^2 ⋮ On sets of type \((m,n)_{r - 1}\) in PG\((r,q)\) ⋮ Generalizing Korchmáros-Mazzocca arcs ⋮ Sets of type-\((1,n)\) in biplanes ⋮ Parameters for sets of type \((m,n)\) in projective planes of prime power order ⋮ Unitals which meet Baer subplanes in 1 modulo \(q\) points ⋮ On the classification of unitals on 28 points of low rank ⋮ Cameron-Liebler line classes with parameter \(x = \frac{q^2 - 1}{2}\) ⋮ A new family of tight sets in \(\mathcal Q^+(5,q)\)

Uses Software

• nauty
• Cayley

Cites Work

• The Steiner systems S(2,4,25) with nontrivial automorphism group
• A 35-set of type (2,5) in PG(2,9)
• Unitary designs on the Hughes plane
• Construction of maximal arcs and dual ovals in translation planes
• A computer search for finite projective planes of order 9
• Some results concerning \(\{(q+1)(n-1);\;n\}\)-arcs and \(\{(q+1)\;(n-1)+1;\;n\}\)-arcs in finite projective planes of order \(q\)
• On subplane partitions of a finite projective plane
• Some new results on sets of type (m,n) in projective planes
• Construction of maximal arcs and partial geometries
• A unital in the Hughes plane of order nine
• Some maximal arcs in finite projective planes
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