ovals in for even have not yet been classified. In an arbitrary finite projective plane of odd order , no sets with more points than , no three of which are collinear, exist, as first pointed out by Bose in a 1947 paper on applications of this sort of mathematics to the statistical design of experiments. Furthermore, by Qvist's theorem, through any point not on an oval there pass either zero or two tangent lines of that oval. When q is even, the situation is completely different. In this case, sets of points, no three of which collinear, may exist in a finite projective plane of order and they are called hyperovals; these are maximal arcs of degree 2. Given an oval there is a unique tangent through each point, and if is even Qvist's theorem, () shows that all these tangents are concurrent in a point outside the oval. Adding this point (called the nucleus of the oval or sometimes the knot) to the oval gives a hyperoval. Conversely, removing any one point from a hyperoval immediately gives an oval. As all ovals in the even order case are contained in hyperovals, a description of the (known) hyperovals implicitly gives all (known) ovals. The ovals obtained by removing a point from a hyperoval are projectively equivalent if and only if the removed points are in the same orbit of the automorphism group of the hyperoval. There are only three small examples (in the Desarguesian planes) where the automorphism group of the hyperoval is transitive on its points (see ) so, in general, there are different types of ovals contained in a single hyperoval. Desarguesian Case: PG(2,2h) This is the most studied case and so the most is known about these hyperovals. Every nonsingular conic in the projective plane, together with its nucleus, forms a hyperoval. These may be called hyperconics, but the more traditional term is regular hyperovals. For each of these sets, there is a system of coordinates such that the set is: However, many other types of hyperovals of PG(2, q) can be found if q > 8. Hyperovals of PG(2, q) for q even have only been classified for q < 64 to date. In PG(2,2h), h > 0, a hyperoval contains at least four points no three of which are collinear. Thus, by the Fundamental Theorem of Projective Geometry we can always assume that the