New (k; r)-arcs in the projective plane of order thirteen

Abstract

A (k;r)-arc $\cal K$ is a set of k points of a projective plane PG(2, q) such that some r, but no r +1 of them, are collinear. The maximum size of a (k; r)-arc in PG(2, q) is denoted by m r (2, q). In this paper a (35; 4)-arc, seven (48; 5)-arcs, a (63; 6)-arc and two (117; 10)-arcs in PG(2, 13) are given. Some were found by means of computer search, whereas the example of a (63; 6)-arc was found by adding points to those of a sextic curve $\cal C$ that was not complete as a (54; 6)-arc. All these arcs are new and improve the lower bounds for m r (2, 13) given in [10, Table 5.4]. The last section concerns the nonexistence of (40; 4)-arcs in PG(2, 13).