A characterization of the sets of internal and external points of a conic

“…The authors wish to thank Dr. Binod Kumar Sahoo for his comments, which helped improve an earlier version of the article. The authors also thank the anonymous referees for their helpful comments and for suggesting to use certain results from the paper [4].…”

Section: Acknowledgmentsmentioning

confidence: 99%

A characterization of the family of secant lines to a hyperbolic quadric in PG(3,q), q odd

Pradhan

Sahu

2020

Discrete Mathematics

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“…There are examples for such sets in AG (2,9). Indeed Penttila and Royle have found in [21] two different sets of type (3, 6) 1 in AG (2,9) with |X| = 36 and automorphism groups of order 2 and 3 (plus the complement of such sets). In [24] Rodgers constructs a set of type (36,45) in AG(2, 81) (plus the complement).…”

Section: The Affine Casementioning

confidence: 99%

On Sets with Few Intersection Numbers in Finite Projective and Affine Spaces

Durante

2014

Electron. J. Combin.

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In this paper we study sets $X$ of points of both affine and projective spaces over the Galois field $\mathop{\rm{GF}}(q)$ such that every line of the geometry that is neither contained in $X$ nor disjoint from $X$ meets the set $X$ in a constant number of points and we determine all such sets. This study has its main motivation in connection with a recent study of neighbour transitive codes in Johnson graphs by Liebler and Praeger [Designs, Codes and Crypt., 2014]. We prove that, up to complements, in $\mathop{\rm{PG}}(n,q)$ such a set $X$ is either a subspace or $n=2,q$ is even and $X$ is a maximal arc of degree $m$. In $\mathop{\rm{AG}}(n,q)$ we show that $X$ is either the union of parallel hyperplanes or a cylinder with base a maximal arc of degree $m$ (or the complement of a maximal arc) or a cylinder with base the projection of a quadric. Finally we show that in the affine case there are examples (different from subspaces or their complements) in $\mathop{\rm{AG}}(n,4)$ and in $\mathop{\rm{AG}}(n,16)$ giving new neighbour transitive codes in Johnson graphs.

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A characterization of the sets of internal and external points of a conic

Cited by 11 publications

References 6 publications

The type of a point and a characterization of the set of external points of a conic inPG(2,q),qodd

The type of a point and a characterization of the set of external points of a conic inPG(2,q),qodd

A characterization of the family of secant lines to a hyperbolic quadric in PG(3,q), q odd

On Sets with Few Intersection Numbers in Finite Projective and Affine Spaces

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