On the Geometry of Affine Difference Sets of Even Order PDF

Keywords

affine difference sets, geometry, even

Abstract

Let A be an affine plane of even order which admits a representation by an affine difference set D in an abelian group G (relative to N), say G = H ⊕ N. We discuss various hyperovals of A related to this representation: (-D + y) ⋃ (∞) is an oval with nucleus y (assuming w.l.o.g. D= 2D), and the sets H + n (n ⋲ N) (which form a partition of A\ (∞)) are ovals with common nucleus ∞. In case A=AG (2, 2°), all these ovals are in fact conics (for which we give explicit equations). In particular, the points of AG (2, 2°)\{(0, 0)} can be partitioned into 2°-1 conics with common nucleus (0, 0); moreover, there are commuting cyclic groups H and N such that H acts regularly on each of these conics whereas N acts regularly on the set of all these conics.