It is known that every ovoid of the parabolic quadric Q(4, q), q = p h , p prime, intersects every three-dimensional elliptic quadric in 1 mod p points. We present a new approach which gives us a second proof of this result and, in the case when p = 2, allows us to prove that every ovoid of Q(4, q) either intersects all the three-dimensional elliptic quadrics in 1 mod 4 points or intersects all the three-dimensional elliptic quadrics in 3 mod 4 points.We also prove that every ovoid of Q(4, q), q prime, is an elliptic quadric. This theorem has several applications, one of which is the non-existence of ovoids of Q(6, q), q prime, q > 3.We conclude with a 1 mod p result for ovoids of Q(6, q), q = p h , p prime.

A particular class of minihypers was studied previously by the authors (in press, Des. Codes Cryptogr.). For q square, this paper improves the results of that work, under the assumption that no weights occur in the minihyper. Using the link between these minihypers and maximal partial s-spreads of PG(N, q), (s+1) | (N+1), the findings on minihypers translate immediately into results on the extendability of partial s-spreads with small positive deficiency. Other applications of this characterisation of minihypers are given by P. Govaerts et al. (in press, European J. Combin.).

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In the first two articles of this series, the structure of certain minihypers was determined. Hamada shows how these results translate into results on linear codes meeting the Griesmer bound, while Govaerts and Storme show how they can be applied to obtain bounds on the size of maximal partial t-spreads and minimal t-covers in finite projective spaces that admit a t-spread.In this article, further applications are given. It is shown that the previously studied minihypers are closely connected to partial t-spreads and t-covers of finite classical polar spaces whose size admits a t-spread.This connection is used to obtain new bounds on the sizes of maximal partial t-spreads of finite classical polar spaces whose sizes admit t-spreads. In order to get a clearer view on which polar spaces these are, divisibility conditions are rewritten into a more convenient form. This yields necessary conditions for the existence of t-spreads in those spaces; it turns out that for some of the polar spaces these conditions are also sufficient.The results on minihypers are then applied to t-covers of the classical polar spaces, and give us a better understanding of their structure.As an immediate corollary to an extendability result for partial t-spreads, a theorem on the extendability of partial ovoids of H (3, q 2 ) is given. This theorem is then used to prove a new upper bound on the size of partial ovoids of H (4, q 2 ), which can be lifted to an upper bound on the size of partial ovoids of H (2n, q 2 ), n ≥ 2. Also partial ovoids on the generalized hexagon H (q) are studied. c 2002 Elsevier Science Ltd. All rights reserved. INTRODUCTORY RESULTSThe thick finite nondegenerate classical polar spaces are:• W 2n+1 (q), the polar space arising from a symplectic polarity of PG(2n + 1, q), n ≥ 1; • Q − (2n + 1, q), the polar space arising from a nonsingular elliptic quadric of PG(2n + 1, q), n ≥ 2;• Q(2n, q), the polar space arising from a nonsingular quadric of PG(2n, q), n ≥ 2;• Q + (2n + 1, q), the polar space arising from a nonsingular hyperbolic quadric of PG(2n + 1, q), n ≥ 2;• H (n, q 2 ), the polar space arising from a nonsingular Hermitian variety in PG(n, q 2 ), n ≥ 3.Let P be a finite classical polar space. A t-spread of P is a set of totally isotropic or singular t-dimensional subspaces that partitions the point set of P. A partial t-spread of P is a set of pairwise disjoint totally isotropic or singular t-dimensional subspaces. It is called maximal when it is not contained in a larger partial t-spread. A t-cover C of P is a set of totally isotropic or singular t-dimensional subspaces such that any point of P is contained in at least one element of C. It is called minimal when it does not contain a smaller t-cover. A generator of P is a maximal totally isotropic or maximal singular subspace of P. The set of all generators of P is denoted by G(P). The rank of P is by definition one more than the dimension of a generator of P. A spread of P is an (r − 1)-spread of P, where r denotes the rank of P.An ovoid O of P is a set of po...

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