Design theory
Affine difference sets of even order
Journal of Combinatorial Theory Series A
Hyperovals in Desarguesian planes: an update
Discrete Mathematics - Special issue on combinatorics
An oval partition of the central units of certain semifield planes
Discrete Mathematics - Special issue on combinatorics
Planar Functions and Planes of Lenz-Barlotti Class II
Designs, Codes and Cryptography
Hyperovals and unitals in Figueroa planes
European Journal of Combinatorics
Two infinite families of failed symmetric designs
Discrete Mathematics - Papers on the occasion of the 65th birthday of Alex Rosa
Polarities and unitals in the Coulter---Matthews planes
Designs, Codes and Cryptography
42-arcs in PG(2, q) left invariant by PSL(2, 7)
Designs, Codes and Cryptography
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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.