Book contents
- Frontmatter
- Contents
- Preface
- Introduction
- Generalized hexagons and BLT-sets
- Orthogonally divergent spreads of Hermitian curves
- Lifts of nuclei in finite projective spaces
- Large minimal blocking sets, strong representative systems, and partial unitals
- The complement of a geometric hyperplane in a generalized polygon is usually connected
- Locally co-Heawood graphs
- A theorem of Parmentier characterizing projective spaces by polarities
- Geometries with diagram (diagram omitted)
- Remarks on finite generalized hexagons and octagons with a point-transitive automorphism group
- Block-transitive t-designs, II: large t
- Generalized Fischer spaces
- Ovoids and windows in finite generalized hexagons
- Flag transitive L.C2 geometries
- On nonics, ovals and codes in Desarguesian planes of even order
- Orbits of arcs in projective spaces
- There exists no (76,21,2,7) strongly regular graph
- Group-arcs of prime power order on cubic curves
- Planar Singer groups with even order multiplier groups
- On a footnote of Tits concerning Dn-geometries
- The structure of the central units of a commutative semifield plane
- Partially sharp subsets of PΓL(n, q)
- Partial ovoids and generalized hexagons
- A census of known flag-transitive extended grids
- Root lattice constructions of ovoids
- Coxeter groups in Coxeter groups
- A local characterization of the graphs of alternating forms
- A local characterization of the graphs of alternating forms and the graphs of quadratic forms over GF(2)
- On some locally 3-transposition graphs
- Coherent configurations derived from quasiregular points in generalized quadrangles
- Veldkamp planes
- The Lyons group has no distance-transitive representation
- Intersection of arcs and normal rational curves in spaces of odd characteristic
- Flocks and partial flocks of the quadratic cone in PG(3, q)
- Some extended generalized hexagons
- Nuclei in finite non-Desarguesian projective planes
On nonics, ovals and codes in Desarguesian planes of even order
Published online by Cambridge University Press: 07 September 2010
- Frontmatter
- Contents
- Preface
- Introduction
- Generalized hexagons and BLT-sets
- Orthogonally divergent spreads of Hermitian curves
- Lifts of nuclei in finite projective spaces
- Large minimal blocking sets, strong representative systems, and partial unitals
- The complement of a geometric hyperplane in a generalized polygon is usually connected
- Locally co-Heawood graphs
- A theorem of Parmentier characterizing projective spaces by polarities
- Geometries with diagram (diagram omitted)
- Remarks on finite generalized hexagons and octagons with a point-transitive automorphism group
- Block-transitive t-designs, II: large t
- Generalized Fischer spaces
- Ovoids and windows in finite generalized hexagons
- Flag transitive L.C2 geometries
- On nonics, ovals and codes in Desarguesian planes of even order
- Orbits of arcs in projective spaces
- There exists no (76,21,2,7) strongly regular graph
- Group-arcs of prime power order on cubic curves
- Planar Singer groups with even order multiplier groups
- On a footnote of Tits concerning Dn-geometries
- The structure of the central units of a commutative semifield plane
- Partially sharp subsets of PΓL(n, q)
- Partial ovoids and generalized hexagons
- A census of known flag-transitive extended grids
- Root lattice constructions of ovoids
- Coxeter groups in Coxeter groups
- A local characterization of the graphs of alternating forms
- A local characterization of the graphs of alternating forms and the graphs of quadratic forms over GF(2)
- On some locally 3-transposition graphs
- Coherent configurations derived from quasiregular points in generalized quadrangles
- Veldkamp planes
- The Lyons group has no distance-transitive representation
- Intersection of arcs and normal rational curves in spaces of odd characteristic
- Flocks and partial flocks of the quadratic cone in PG(3, q)
- Some extended generalized hexagons
- Nuclei in finite non-Desarguesian projective planes
Summary
Abstract
A nonic in PG(2, q), q = 2h, is defined to be either a non-degenerate conic plus its nucleus, or a degenerate conic, different from a repeated line, minus its nucleus. It is noted that every nonic is in the dual-line code of the plane, and so several questions arise. Can we have collections of nonics generating this code? Each oval (or (q + 2)-arc) is the sum (mod 2) of various numbers of nonics – what is the minimum number?
Introduction
There are four kinds of conics in a projective plane π := PG(2, q) over a finite field GF(q); see [5]. These are:
(1) irreducible conic, having precisely q + 1 points;
(2) distinct lines, having 2q + 1 points;
(3) lines in the quadratic extension PG(2, q2), intersecting in a point of π, and so having only one point;
(4) repeated line.
When q is even (and so q = 2h) the first three types of conies define a certain point called the nucleus, which lies on the conic if and only if the conic is reducible. In the first case it is the point of intersection of all the tangents; in the second it is the intersection of the two lines; in the third case the nucleus is the only point of π that is on the conic.
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- Finite Geometries and Combinatorics , pp. 155 - 160Publisher: Cambridge University PressPrint publication year: 1993