Arcs and Ovals from Abelian Groups

Abstract

We use large abelian collineation groups of finite projective planes (via the associated representation by some sort of difference set) to construct interesting families of (hyper)ovals. This approach is of particular interest for groups of type (b) in the Dembowski-Piper classification, i.e., for abelian relative (n, n, n, 1)-difference sets. Here we obtain the first series of ovals in planes of Lenz-Barlotti class II.1, namely in the Coulter-Matthews planes, and a partition of the affine part of any commutative semifield plane of even order into translation ovals; we also provide a somewhat surprising embedding of the dual affine translation plane into the original projective plane and an explicit description of a maximal arc of degree q/2 which leads to the embedding of a certain Hadamard design as a family of maximal arcs. We also survey previous results for groups of type (a) and (d), i.e., for planar and affine difference sets, respectively. Finally, we study the case of groups of type (f) (which correspond to direct product difference sets); here we also use the resulting ovals to give considerably simpler proofs for some known restrictions concerning planes admitting such a group.