On the maximum size of some (k,r)(k,r)-arcs in PG(2,q)PG(2,q)

A (k,r)(k,r)-arc is a set of k points of a projective plane such that some r  , but no r+1r+1 of them, are collinear. The maximum size of a (k,r)(k,r)-arc in PG(2,q)PG(2,q) is denoted by mr(2,q)mr(2,q). In this paper we prove that mr(2,q)⩽(r-1)q+r-(q+3)/2mr(2,q)⩽(r-1)q+r-(q+3)/2 for r>(q+3)/2r>(q+3)/2 and q=17,19,23,29.q=17,19,23,29. As a consequence the nonexistence of 34 three-dimensional codes over GF(q),GF(q),q=17,19,23,29q=17,19,23,29, is proved.