A characterization of translation ovals in finite even order planes

In this article we consider a set C of points in PG ( 4 , q ) , q even, satisfying certain combinatorial properties with respect to the planes of PG ( 4 , q ) . We show that there is a regular spread in the hyperplane at infinity, such that in the corresponding Bruck-Bose plane PG ( 2 , q 2 ) , the points corresponding to C form a translation hyperoval, and conversely.