On small dense arcs in Galois planes of square order

AbstractIn 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 (4q−4,2)-arcs for q even, and we also show that they are complete as far as q⩽1024