A note on Pall partitions over finite fields

A Pall partition for a quadratic space V is a collection of disjoint (except for {0}) maximal totally isotropic subspaces whose union contains all of the isotropic vectors in V. In this paper it is shown that no non-degenerate quadratic space of dimension 4k+1, k?1, over a finite field of odd characteristic can have a Pall partition. The method of proof consists of assuming such a partition exists and showing by various counting arguments that this leads to the existence of an impossible array of ordered pairs.