Ovals and hyperovals in nets

We prove that nets of order n with small deficiency d relative to n contain no hyperovals unless n is even and d⩽2. Secondly, we examine the problems of the existence of r-nets of order n⩽8 with ovals or hyperovals; we are able to reduce these problems to a finite number of undetermined orders n. Thirdly, we prove the existence of a set of 7 incomplete mutually orthogonal Latin squares of order n with a hole of size 8 for every integer n⩾775. As a corollary, there exists a 9-net of order n with a hyperoval for every n⩾775.