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,n)-arcs and the falsity of the Lunelli-Sce conjecture
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
INTRODUCTION
A (k,n)-arc in PG(2,q) is a set of k points, such that some n, but no n+1, of them are collinear. The maximum value of k for which a (k,n)-arc exists in PG(2,q) will be denoted by m (n)2,q - Clearly m(l)2,q = 1 and m(q+l)2,q = q2 + q + 1, and so from now on we assume that 2 ≤ n ≤ q.
The following two theorems are well-known, and proofs may be found in Chapter 12 of Hirschfeld [12].
THEOREM 1.1:
(i) m(n)2,q ≤ (n-l)q + n.
(ii) (Cossu [8]). For n≤q, equality can occur in (i) only if n | q.
A ((n-l)q + n, n)-arc is known as a maximal arc.
THEOREM 1.2: There exists a maximal arc in PG(2,q) when
(i) n = q, for any q, a maximal arc being the complement of a line,
(ii) (Denniston [10]). q = 2h, and n is any divisor of q.
It is not known whether maximal arcs exist when q is odd and 2 ≤ n < q, although it has been proved by Thas [16] that there are no (2q+3, 3)- arcs in PG(2, 3h) for h > 1.
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- Finite Geometries and DesignsProceedings of the Second Isle of Thorns Conference 1980, pp. 153 - 168Publisher: Cambridge University PressPrint publication year: 1981
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