On large maximal partial ovoids of the parabolic quadric Q(4, q)

Abstract

We use the representation T 2 (Ø) for Q(4, q) to show that maximal partial ovoids of Q(4, q) of size q 2 − 1, q = p h , p odd prime, h > 1, do not exist. Although this was known before, we give a slightly alternative proof, also resulting in more combinatorial information of the known examples for q odd prime. keywords: maximal partial ovoid, generalized quadrangle, parabolic quadric.MSC (2010): 05B25, 51D20, 51E12, 51E20, 51E21.

On the structure of the directions not determined by a large affine point set

“…These examples occur in the theory of maximal partial ovoids of generalized quadrangles, and where studied in [12], [4], and [6]. Non-existence of such examples for q = p h , p an odd prime, h > 1, was shown in [7]. Now we prove a general extendability theorem in the 3-space if ε < p. Proof.…”

“…We consider the case when U is extendible as the typical one: otherwise N has a very restricted (strong) structure; although note that there exist examples of maximal point sets U , of size q 2 − 2, q ∈ {3, 5, 7, 11}, not determining the points of a conic at infinity. These examples occur in the theory of maximal partial ovoids of generalized quadrangles, and where studied in [12], [4], and [6]. Non-existence of such examples for q = p h , p an odd prime, h > 1, was shown in [7].…”

“…An extendability result known for general dimension is the following. Originally, it was proved in [7] for n = 3. A proof for general n can be found in [1].…”

The Known Maximal Partial Ovoids of Size of

“…Yet another alternative is to employ a computer to at least tackle the smaller cases. We have proved by computer that for and 11, the affine ovoids are indeed unique and that for none exist (which confirms the result of ). For our program takes only a few minutes of CPU time while for already 3 weeks were needed.…”

“…In this paper, we shall be concerned with the next case, that of maximal partial ovoids of size . It is shown in that maximal partial ovoids of of size do not exist when q is odd and not prime. When q is odd and prime, examples of maximal partial ovoids of this size are known for and 11, but none for , .…”