Designs, Codes and Cryptography

, Volume 80, Issue 2, pp 317–332 | Cite as

Characterising pointsets in \(\mathrm{{PG}}(4,q)\) that correspond to conics

  • S. G. BarwickEmail author
  • Wen-Ai Jackson


We consider a non-degenerate conic in \(\mathrm{{PG}}(2,q^2)\), \(q\) odd, that is tangent to \(\ell _\infty \) and look at its structure in the Bruck–Bose representation in \(\mathrm{{PG}}(4,q)\). We determine which combinatorial properties of this set of points in \(\mathrm{{PG}}(4,q)\) are needed to reconstruct the conic in \(\mathrm{{PG}}(2,q^2)\). That is, we define a set \({\mathcal {C}}\) in \(\mathrm{{PG}}(4,q)\) with \(q^2\) points that satisfies certain combinatorial properties. We then show that if \(q\ge 7\), we can use \({\mathcal {C}}\) to construct a regular spread \({\mathcal {S}}\) in the hyperplane at infinity of \(\mathrm{{PG}}(4,q)\), and that \({\mathcal {C}}\) corresponds to a conic in the Desarguesian plane \(\mathcal {P}({\mathcal {S}})\cong \mathrm{{PG}}(2,q^2)\) constructed via the Bruck–Bose correspondence.


Finite projective geometry Bruck–Bose representation Conics  Characterization 

Mathematics Subject Classification



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Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  1. 1.School of MathematicsUniversity of AdelaideAdelaideAustralia

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