Book contents
- Frontmatter
- Contents
- PREFACE
- Introduction
- Generalized Steiner systems of type 3-(v, {4,6}, 1)
- Some remarks on D.R. Hughes' construction of M12 and its associated designs
- On k-sets of class [0,1,2,n]2 in PG(r,q)
- Covering graphs and symmetric designs
- Arcs and blocking sets
- Flat embeddings of near 2n-gons
- Codes, caps and linear spaces
- Geometries originating from certain distance-regular graphs
- Transitive automorphism groups of finite quasifields
- On k-sets of type (m,n) in projective planes of square order
- On k-sets of type (m,n) in a Steiner system S(2, l, v)
- Some translation planes of order 81
- A new partial geometry constructed from the Hoffman-Singleton graph
- Locally cotriangular graphs
- Coding theory of designs
- On shears in fixed-point-free affine groups
- On (k,n)-arcs and the falsity of the Lunelli-Sce conjecture
- Cubic surfaces whose points all lie on their 27 lines
- Existence results for translation nets
- Translation planes having PSL(2,w) or SL(3,w) as a collineation group
- Sequenceable groups: a survey
- Polar spaces embedded in a projective space
- On relations among the projective geometry codes
- Partition loops and affine geometries
- Regular cliques in graphs and special 1½ designs
- Bericht über Hecke Algebren und Coxeter Algebren eindlicher Geometrien
- On buildings and locally finite Tits geometries
- Moufang conditions for finite generalized quadrangles
- Embedding geometric lattices in a projective space
- Coverings of certain finite geometries
- On class-regular projective Hjelmslev planes
- On multiplicity-free permutation representations
- On a characterization of the Grassmann manifold representing the lines in a projective space
- Affine subplanes of projective planes
- Point stable designs
- Other talks
- Participants
On k-sets of type (m,n) in projective planes of square order
Published online by Cambridge University Press: 05 April 2013
- Frontmatter
- Contents
- PREFACE
- Introduction
- Generalized Steiner systems of type 3-(v, {4,6}, 1)
- Some remarks on D.R. Hughes' construction of M12 and its associated designs
- On k-sets of class [0,1,2,n]2 in PG(r,q)
- Covering graphs and symmetric designs
- Arcs and blocking sets
- Flat embeddings of near 2n-gons
- Codes, caps and linear spaces
- Geometries originating from certain distance-regular graphs
- Transitive automorphism groups of finite quasifields
- On k-sets of type (m,n) in projective planes of square order
- On k-sets of type (m,n) in a Steiner system S(2, l, v)
- Some translation planes of order 81
- A new partial geometry constructed from the Hoffman-Singleton graph
- Locally cotriangular graphs
- Coding theory of designs
- On shears in fixed-point-free affine groups
- On (k,n)-arcs and the falsity of the Lunelli-Sce conjecture
- Cubic surfaces whose points all lie on their 27 lines
- Existence results for translation nets
- Translation planes having PSL(2,w) or SL(3,w) as a collineation group
- Sequenceable groups: a survey
- Polar spaces embedded in a projective space
- On relations among the projective geometry codes
- Partition loops and affine geometries
- Regular cliques in graphs and special 1½ designs
- Bericht über Hecke Algebren und Coxeter Algebren eindlicher Geometrien
- On buildings and locally finite Tits geometries
- Moufang conditions for finite generalized quadrangles
- Embedding geometric lattices in a projective space
- Coverings of certain finite geometries
- On class-regular projective Hjelmslev planes
- On multiplicity-free permutation representations
- On a characterization of the Grassmann manifold representing the lines in a projective space
- Affine subplanes of projective planes
- Point stable designs
- Other talks
- Participants
Summary
Let π be a projective plane of order q and let K be a subset of its point-set, |K| = k; the set K will be called a two character k-set of type (m,n) (or briefly a k-set of type (m,n)) if a line in π either meets K in m or in n points and m-secants and n-secants actually exist; hence 0 ≤ m < n ≤ q + l.
In 1966, Tallini Scafati [6] gave necessary arithmetical conditions for a k-set of type (m,n) to exist in a projective plane of order q, q = ph, p a prime, h a non-negative integer. Namely, she proved that k must be a root of the equation
k2 - k(n + m + q(n + m - 1)) + mn(q2 + q+1) = 0, (1)
so that its discriminant must be a non-negative square; moreover, n-m must divide q. As a special case, she characterized k-sets of type (l,n), n ≠ q + 1, proving that q must be a square, n = √q +1 and such an arc either is a Baer subplane or a Hermitian arc (a Hermitian curve if a certain reciprocity condition holds). Thus sets of type (m,q), m ≠ 0, are also characterized, being the complements of the preceding ones. A set of type (1,q+1) is a line; a set of type (0,q) is an affine plane.
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- Finite Geometries and DesignsProceedings of the Second Isle of Thorns Conference 1980, pp. 98 - 103Publisher: Cambridge University PressPrint publication year: 1981
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