On small dense arcs in Galois planes of square order

In the Galois projective plane of square order q, we show the existence of small dense (k,4)-arcs whose points lie on two conics for q odd, and on two hyperovals for q even. We provide an explicit construction of (4rootq - 4,2)-arcs for q even, and we also show that they are complete as far as q less than or equal to 1024. (C) 2002 Elsevier Science B.V. All rights reserved.