Hyperovals in the Translation Planes of Order \(16\)

The translation planes of order 16 have been classified by Dempwolff and Reifart \([4]\). Using this classification, and in particular the spreads given in that paper, we have conducted a complete computer search for the hyperovals (18-arcs) in each of these planes. With few exceptions, the hyperovals obtained are hyperbolic (having two points on the special line at infinity) and are of a type we call translation hyperovals. The only exceptions occur in the plane over the semifield with kernel \({GF}(2)\). In this plane there also appear a class of elliptic (having no points on the special line at infinity) hyperovals and two classes of hyperbolic hyperovals which are not translation hyperovals. The automorphism groups of the hyperovals are also determined.