A new hyperoval in \(PG(2,32)\)
A \(k\)-arc in a projective plane is a set of \(k\) points, no three of which are collinear. An \((n+1)\)-arc in a projective plane of order \(n\) is an oval. When \(n\) is odd, there are no \((n+2)\)-arcs. When \(n\) is odd and the plane is Desarguesian, a well known theorem of Segre says that the oval is a conic. When \(n\) is even, an oval can be extended to a hyperoval i.e., an \((n+2)\)-arc. The authors classify the hyperovals in the Desarguesian plane of order 32 and give one new one. The approach is to represent the hyperovals as sets of points in homogeneous coordinates of the form \(\{(1,t,f(t)\}\cup\{(0,1,0),(0,0,1)\}\) when \(f\) is a permutation polynomial over \(GF(32)\). The author gives the automorphism groups of these hyperovals.
scientific article; zbMATH DE number 92579
scientific article