Classification of ovoids in \(PG(3,32)\)
An ovoid in the projective space \(\text{PG}(3,q)\) is a set of \(a^ 2+1\) points, no three of which are collinear. For each prime power \(q\), an elliptic quadric provides an example which is the only possibility for odd \(q\). For even \(q\) the only other known examples is the Tits ovoid which exists only for nonsquare \(q>4\). Until recently a classification of ovoids in spaces of even order had been completed only for \(q=2\), 4 and 8. In general the classification of ovoids in \(\text{PG} (3,q)\) seems to depend on a classification of the ovals in \(\text{PG} (2,q)\), a very difficult problem in itself. However, both problems have been solved for \(q=16\), so the next case to consider is \(q=32\). Very recently the ovals in \(\text{PG} (2,32)\) were classified by \textit{T. Penttila} and \textit{G. F. Royle} [ibid., 151-158 (1994; see the paper below)], thus enabling the present authors to complete the ovoid classification problem for \(q=32\). An interesting mixture of theory and computer computation is required, with the end result being that the only ovoids of PG(3, 32) are the two well known examples. Hence by a famous result of P. Dembowski on inversive planes of even order, there are exactly two inverse planes of order 32.
scientific article; zbMATH DE number 621997