Small complete arcs in André planes of square order

AbstractA complete are in a projective plane of orderq has at least $$\sqrt {2q} $$ points. We show the existence of completek-arcs having $$k< C\sqrt q \log ^2 q$$ points in certain André-planes of square order. Moreover our construction shows that for all $$x,q > x > \sqrt q \log q$$ there are completek-arcs withx