On a Particular Class of Minihypers and its Applications. III. Applications

Abstract

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...

Partial ovoids and partial spreads in hermitian polar spaces

“…It is known that H(4, q 2 ) has no ovoids [17]. An upper bound on the size of partial ovoids in H(4, q 2 ) is known [11]. We will improve on this result.…”

“…The uniqueness of this extension to a spread follows from the fact that S is extended by less than q + 1 lines to a spread. (b) See [11].…”

“…Then research was focused on the size of the largest partial ovoids and spreads of finite classical polar spaces that do not have ovoids or spreads, and to the problem of the extendability of partial ovoids and partial spreads to ovoids and spreads when the finite classical polar spaces have ovoids and spreads [11,18].…”

Tight sets, weighted m-covers, weighted m-ovoids, and minihypers

“…Recently, many results in finite geometry were obtained by applying characterization results on minihypers [8,17,18]. In this paper, using characterization results on certain minihypers, we present new results on tight sets in classical finite polar spaces and weighted m-covers, and on weighted m-ovoids of classical finite generalized quadrangles.…”

“…Although minihypers were first introduced to study the problem of linear codes meeting the Griesmer bound [20,21], characterization results on minihypers can be used to solve problems in finite geometry, see [8], [18] and [4] for applications on substructures of finite projective spaces and generalized quadrangles. We refer to [33] for a survey on the use of minihypers in the study of linear codes meeting the Griesmer bound and in the study of geometrical problems.…”