A sporadic ovoid in Omega+(8, 5) and some non-desarguesian translation planes of order 25

An ovoid is an orthogonal vector space V of type .Q + (2n, q) is a set 0 of q-1 + 1 pairwise non-orthogonal singular points (one-spaces). Every maximal singular subspace of V will contain a unique point of 0, so it is extremal. Ovoids have connections to coding theory, and translation planes (cf. Kantor [2]). There are no known examples for n 2 5 and there is evidence that they will not exist (see Kantor [2], Shult [6], and Thas [7]). Examples for n = 4 are rare, though Kantor has constructed several families for certain prime powers q, Conway, Kleidman, and Parker [l] have established existence for all primes q, and Shult [S] has given an interesting example for q = 7. When I? = 3, ovoids are equivalent to affine translation planes, via the Klein correspondence as described by Mason and Shult 1143. In this note we explicitly construct an ovoid in Sz + (8, 5). We determine its full stabilizer, Z; and show it contains the symmetric group on ten letters, S,,. Additionally, we will compute the orbits of the stabilizer on all the other singular points of V. This will enable us to construct three ovoids in 52+(6, 5), and hence three afline translation planes. It will be shown that these planes are non-Desarguesian. One of the planes is the Hering plane of order 25 admitting S&(9).