Patrick Govaerts

On Ovoids of Parabolic Quadrics

It is known that every ovoid of the parabolic quadric Q(4, q), q=ph, 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=ph, p prime.

On a Particular Class of Minihypers and Its Applications

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.).

On a Particular Class of Minihypers and Its Applications. I. The Result for General q

This article is the first in a series of three articles that discuss a particular class of minihypers and its applications. Proving that for small δ and μ < N, a {δvμ + 1, δvμ; N, q}-minihyper consists of a sum of δ μ-spaces, we show that the excess points of an s-cover with excess δ of PG(N, q), (s + 1)|(N + 1), form a sum of δ s-spaces, and that no maximal partial s-spreads with deficiency δ of PG(N, q), (s + 1)|(N + 1), exist. The case q square will be studied in greater detail in [7] and further applications of these classification results on this class of minihypers will be published in [8].

Cameron-Liebler line classes in PG(3,4)

A Cameron-Liebler line class is a set L of lines in PG(3, q) for which there exists a number x such that |L⋂S|=x for all spreads S. There are many equivalent properties: Theorem 1 lists eight. This paper classifies Cameron-Liebler line classes with x⩽4 (with two exceptions). It is also shown that the study of Cameron-Liebler line classes is equivalent to the study of weighted sets of points in PG(3, q) with two weights on lines.

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

Minihypers are substructures of projective spaces introduced to study linear codes meeting the Griesmer bound. Recently, many results in finite geometry were obtained by applying characterization results on minihypers (De Beule et al. 16:342–349, 2008; Govaerts and Storme 4:279–286, 2004; Govaerts et al. 28:659–672, 2002). 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. The link with minihypers gives us characterization results of i-tight sets in terms of generators and Baer subgeometries contained in the Hermitian and symplectic polar spaces, and in terms of generators for the quadratic polar spaces. We also present extendability results on partial weighted m-ovoids and partial weighted m-covers, having small deficiency, to weighted m-covers and weighted m-ovoids of classical finite generalized quadrangles. As a particular application, we prove in an alternative way the extendability of 53-, 54-, and 55-caps of PG(5,3), contained in a non-singular elliptic quadric Q−(5,3), to 56-caps contained in this elliptic quadric Q−(5,3).

The classification of the smallest nontrivial blocking sets in PG( n , 2)

We determine the smallest nontrivial blocking sets with respect to t-spaces in PG(n, 2), n ≥ 3. For t = n - 1, they are skeletons of solids in PG(n, 2); for 1 ≤ t < n - 1, they are cones with vertex an (n - t - 3)- space πn-t - 3 and base the set of points on the edges of a tetrahedron in a solid skew to πn-t-3.

Some New Maximal Sets of Mutually Orthogonal Latin Squares

Two ways of constructing maximal sets of mutually orthogonal Latin squares are presented.The first construction uses maximal partial spreads in PG(3, 4) \ PG(3, 2) with r lines, where r ∈ {6, 7}, to construct transversal-free translation nets of order 16 and degree r + 3 and hence maximal sets of r + 1 mutually orthogonal Latin squares of order 16. Thus sets of t MAXMOLS(16) are obtained for two previously open cases, namely for t = 7 and t = 8.The second one uses the (non)existence of spreads and ovoids of hyperbolic quadrics Q+ (2m + 1, q), and yields infinite classes of q2n − 1 − 1 MAXMOLS(q2n), for n ≥ 2 and q a power of two, and for n = 2 and q a power of three.