Innovations in Incidence Geometry

On sharply transitive sets in $\mathsf{PG}(2,q)$

Alexander A. Davydov, Massimo Giulietti, Stefano Marcugini, and Fernanda Pambianco

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In PG ( 2 , q ) a point set K is sharply transitive if the collineation group preserving K has a subgroup acting on K as a sharply transitive permutation group. By a result of Korchmáros, sharply transitive hyperovals only exist for a few values of q , namely q = 2 , 4 and 1 6 . In general, sharply transitive complete arcs of even size in PG ( 2 , q ) with q even seem to be sporadic. In this paper, we construct sharply transitive complete 6 ( q 1 ) -arcs for q = 4 2 h + 1 , h 4 . As far as we are concerned, these are the smallest known complete arcs in PG ( 2 , 4 7 ) and in PG ( 2 , 4 9 ) ; also, 4 2 seems to be a new value of the spectrum of the sizes of complete arcs in PG ( 2 , 4 3 ) . Our construction applies to any q which is an odd power of 4 , but the problem of the completeness of the resulting sharply transitive arc remains open for q 4 1 1 . In the second part of this paper, sharply transitive subsets arising as orbits under a Singer subgroup are considered and their characters, that is the possible intersection numbers with lines, are investigated. Subsets of PG ( 2 , q ) and certain linear codes are strongly related and the above results from the point of view of coding theory will also be discussed.

Article information

Innov. Incidence Geom., Volume 6-7, Number 1 (2007), 139-151.

Received: 29 February 2008
Accepted: 28 April 2008
First available in Project Euclid: 28 February 2019

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 51E21: Blocking sets, ovals, k-arcs

complete arcs transitive arcs intersection numbers


Davydov, Alexander A.; Giulietti, Massimo; Marcugini, Stefano; Pambianco, Fernanda. On sharply transitive sets in $\mathsf{PG}(2,q)$. Innov. Incidence Geom. 6-7 (2007), no. 1, 139--151. doi:10.2140/iig.2008.6.139.

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