found the hyperoval: f(x) = x4 + x8 + x14 + x34 + x42 + x48 + x62 + η21(x6+x16 +x26+x28+x30+x32+x40+x58) + η42(x10 + x18 + x24 + x36 + x44 + x50 + x52+ x60), which has an automorphism group of order 12 (η is a primitive element of GF(64) as above). This hyperoval is the first distinct Adelaide hyperoval. Penttila and Royle have shown that any other hyperoval in this plane would have to have a trivial automorphism group. This would mean that there would be many projectively equivalent copies of such a hyperoval, but general searches to date have found none, giving credence to the conjecture that there are no others in this plane. Abstract ovals Following (Bue1966), an abstract oval, also called a B-oval, of order is a pair where is a set of elements, called points, and is a set of involutions acting on in a sharply quasi 2-transitive way, that is, for any two with for , there exists exactly one with and . Any oval embedded in a projective plane of order might be endowed with a structure of an abstract oval of the same order. The converse is, in general, not true for ; indeed, for there are two abstract ovals which may not be embedded in a projective plane, see (Fa1984). When is even, a similar construction yields abstract hyperovals, see (Po1997): an abstract hyperoval of order is a pair where is a set of elements and is a set of fixed-point free involutions acting on such that for any set of four distinct elements there is exactly one with . See also Ovoid (projective geometry) Notes References External links Bill Cherowitzo's Hyperoval Page Projective geometry Incidence geometry