A sporadic ovoid in \(\Omega ^ +(8,5)\) and some non-Desarguesian translation planes of order 25 (Q919606)

An ovoid in an orthogonal vector space V of type \(\Omega^+(2n,q)\) is a set 0 of \(q^{n-1}+1\) pairwise non-orthogonal singular points (one- spaces). The author constructs an ovoid in \(\Omega^+(8,5)\). He determines its full stabilizer \(\Sigma\) and shows that it contains the symmetric group on ten letters, \(S_{10}.\) He computes the orbits of the stabilizer on all the other singular points of V. This enables him to construct three ovoids in \(\Omega^+(6,5)\) and hence three affine translation planes. It is shown that these planes are non-Desarguesian. One of the plane is the Hering plane of order 25. The avoid is constructed on the basis of the linear 10-dimensional space with orthogonality over the field GF(5).