Doubleoval

Doubleoval is a concept in finite geometry that extends the notion of an oval in a projective plane. An oval in a projective plane of order q is a set of q + 1 points with no three collinear. A doubleoval replaces the single oval with a set of 2q points, still no three collinear, but with the additional property that every line of the plane meets the set in 0, 2, or 4 points. This definition ensures that the combinatorial structure of the doubleoval is symmetric and regular, and makes it a maximal arc of degree 2 in the plane.

The study of doubleovals began in the 1950s and 1960s by mathematicians such as Segre and Hirschfeld, who investigated their existence and classification in planes of even order. For odd orders, a doubleoval can only exist as the union of two disjoint ovals, but such unions do not satisfy the 0, 2, 4 incidence condition, and hence are not considered true doubleovals. In even order planes, non‑degenerate doubleovals exist and are closely tied to Denniston’s maximal arcs, and there is a known correspondence between doubleovals and certain quadratic forms over finite fields.

Applications of doubleovals appear in coding theory and combinatorial design. Because the incidence matrix of a doubleoval has regular row and column sums, it can be used to construct symmetric block designs and two‑weight codes with desirable error‑correcting properties. Moreover, the combinatorial geometry of doubleovals informs the study of blocking sets and can be used to bound the sizes of Kakeya sets in finite geometries.

Open research questions remain regarding the complete classification of doubleovals in all finite projective planes. While complete classifications exist for planes of small even order, the existence of doubleovals in larger even orders and their possible symmetries continue to be investigated.