Arcs with Large Conical Subsets

Abstract

We classify the arcs in $\mathrm{PG}(2,q)$, $q$ odd, which consist of $(q+3)/2$ points of a conic $C$ and two points not on te conic but external to $C$, or $(q+1)/2$ points of $C$ and two additional points, at least one of which is an internal point of $C$. We prove that for arcs of the latter type, the number of points internal to $C$ can be at most $4$, and we give a complete classification of all arcs that attain this bound. Finally, we list some computer results on extending arcs of both types with further points.

The Complete Arcs of PG(2,31)

“…For q = 31 we find the unique complete arc of size 20 with G S D 20 .2 and for q = 67 we find a complete arc of size 44 with G S D 44 .2. The latter is larger than the arcs of size 36 or 37 that can be obtained by extending a large conical subset with external or internal points [6] and also falls outside the range of known complete arcs given in [ Finally, note that the recipe (and the arguments) of this section can be extended to kth powers instead of cubes, yielding arcs of sizes 2 k (q − 1), 2 k (q − 1) + 1 and 2 k (q − 1) + 2. Unfortunately, for k > 3, the resulting arcs are rather small.…”

Arcs with Large Conical Subsets in Desarguesian Planes of Even Order

“…(Where possible we have chosen notations in such a way that they conform to the notations for similar concepts in [1]. )…”

“…As in [1], apart from the 'rotations' M i defined in (6) we also introduce the 'symmetries' M i , as follows:…”

“…After these preliminaries we can continue with the classification up to projective equivalence of arcs with large conical subsets much in the same way as when q is odd. Because of the similarity with [1] we shall try to be brief. First q must be large enough to ensure that the conic C is completely determined by the arc S. Let S be an arc with conical subset R and excess |S \ R| = e. If C is another conic giving rise to the conical subset R = S ∩ C with |S \ R | = e .…”

“…In [1] we classified all arcs of this type with two points not on the conic, for odd q only. Our method made use of an initial step that needs an element of the field GF(q) that is not a square.…”