A full classification (up to equivalence) of all complete k-arcs in the Desarguesian projective planes of order 27 and 29 was obtained by computer. The resulting numbers of complete arcs are tabulated according to size of the arc and type of the automorphism group, and also according to the type of algebraic curve into which they can be embedded. For the arcs with the larger automorphism groups, explicit descriptions are given.The algorithm used for generating the arcs is an application of isomorphfree backtracking using canonical augmentation, an adaptation of an earlier algorithm by the authors.Part of the computer results can be generalized to other values of q: two families of arcs are presented (of size 12 and size 18) for which the symmetric group S 4 is a group of automorphisms.
We have classified by computer the projectively distinct complete (k, 3)-arcs in PG(2, q), q ≤ 13. The algorithm used is an application of isomorphfree backtracking using canonical augmentation, an adaptation of our earlier algorithms for the generation of (k, 2)-arcs. We describe those parts of the algorithms which are specific to the particular problem of (k, 3)-arcs. For each of these arcs we have also determined the automorphism group. The results are summarised in tables where the arcs are listed according to size and automorphism group. For the arcs with the larger automorphism groups, explicit descriptions are given. Part of the computer results can be generalized to other values of q: we describe constructions of arcs having S 4 as a group of automorphisms, arcs containing the union of three 'half conics' and arcs constructed from parts of cubic curves.
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.
A full classification (up to equivalence) of all complete k-arcs in the Desarguesian projective planes of order 23 and 25 was obtained by computer. The algorithm used is an application of isomorph-free backtracking using canonical augmentation, as introduced by McKay, which we have adapted to the case of subset generation in Desarguesian projective planes. We have applied two variants of the same algorithm, and both techniques yield exactly the same results. Earlier (partial) results by other authors on k-arcs in PG(2, q) with q ≤ 25, are reproduced by our programs. We describe those parts of the algorithms which are relevant to the particular problem of generating k-arcs and which have made this project feasible. We also list the number of complete arcs in PG(2, 23) and PG(2, 25) according to size of the arc and type of the automorphism group. Explicit descriptions are given for the arcs with the larger automorphism groups. q
We give an explicit classification of the arcs in PG$(2,q)$ ($q$ even) with a large conical subset and excess 2, i.e., that consist of $q/2+1$ points of a conic and two points not on that conic. Apart from the initial setup, the methods used are similar to those for the case of odd $q$, published earlier (Electronic Journal of Combinatorics, 17, #R112).
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