On sharply transitive sets in PG(2,q)

Abstract

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 permuta- tion group. By a result of Korchmsharply transitive hyperovals only exist for a few values of q, namely q = 2, 4 and 16. In general, sharply transitive complete arcs of even size in PG(2, q) with q even seem to be spo- radic. In this paper, we construct sharply transitive complete 6( p q 1)-arcs for q = 4 2h+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, 42 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 11 . 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.