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, q2) is given. This theorem is then used to prove a new upper bound on the size of partial ovoids of H(4, q2), which can be lifted to an upper bound on the size of partial ovoids of H(2 n, q2),n≥ 2. Also partial ovoids on the generalized hexagon H(q) are studied.