Maximal partial ovoids and maximal partial spreads in hermitian generalized quadrangles

Maximal partial ovoids and maximal partial spreads of the hermitian generalized quadrangles H(3,q²) and H(4,q²) are studied in great detail. We present improved lower bounds on the size of maximal partial ovoids and maximal partial spreads in the hermitian quadrangle H(4,q²). We also construct in H(3,q²), q=2^2h+1, h≥ 1, maximal partial spreads of size smaller than the size q²+1 presently known. As a final result, we present a discrete spectrum result for the deficiencies of maximal partial spreads of H(4,q²) of small positive deficiency δ. © 2007 Wiley Periodicals, Inc. J Combin Designs 16: 101–116, 2008     

Ovoids and translation planes revisited

Kantor has previously described the translation planes which may be obtained by projecting sections of ovoids in ⁺(8, q)-spaces to ovoids in corresponding ⁺(6, q)-spaces. Since the Klein correspondence associates spreads in 4-dimensional vector spaces with ovoids in ⁺(6, q)-spaces, there are corresponding translation planes of order q ² and kernel containing GF(q). In this article, we revisit some of these translation planes and give some presentations of the spreads. Motivated by various properties of the planes, we study, in general, translation planes which admit certain homology groups and/or elation groups. In particular, we develop new constructions of projective planes of Lenz-Barlotti class II-1.Finally, we show how certain projective planes of order q ² of Lenz-Barlotti class II-1 may be considered equivalent to flocks of quadratic cones in PG(3, q).This work was partially supported by NSF grant DMS-8800843.     

Spreads and ovoids in finite generalized quadrangles

This paper is a survey on the existence and non-existence of ovoids and spreads in the known finite generalized quadrangles. It also contains the following new results. We prove that translation generalized quadrangles of order (s,s ²), satisfying certain properties, have a spread. This applies to three known infinite classes of translation generalized quadrangles. Further a new class of ovoids in the classical generalized quadranglesQ(4, 3 ^e ),e3, is constructed. Then, by the duality betweenQ(4, 3 ^e ) and the classical generalized quadrangleW (3 ^e ), we get line spreads of PG(3, 3 ^e ) and hence translation planes of order 3^2e . These planes appear to be new. Note also that only a few classes of ovoids ofQ(4,q) are known. Next we prove that each generalized quadrangle of order (q ²,q) arising from a flock of a quadratic cone has an ovoid. Finally, we give the following characterization of the classical generalized quadranglesQ(5,q): IfS is a generalized quadrangle of order (q,q ²),q even, having a subquadrangleS isomorphic toQ(4,q) and if inS each ovoid consisting of all points collinear with a given pointx ofS\S is an elliptic quadric, thenS is isomorphic toQ(5,q).     

Ovoids from theE 8 root lattice

New families of ovoids are constructed in O ₈ ⁺ (p) forp prime, using theE ₈ root lattice, generalizing a construction of Conway, Kleidman and Wilson. Using this construction, it appears likely that O ₈ ⁺ (p) has unboundedly many ovoids asp.     

Partial ovoids and partial spreads in hermitian polar spaces

We present improved lower bounds on the sizes of small maximal partial ovoids in the classical hermitian polar spaces, and improved upper bounds on the sizes of large maximal partial spreads in the classical hermitian polar spaces. Of particular importance is the presented upper bound on the size of a maximal partial spread of H(3,q ²). For q = 2,3, the presented upper bound is sharp. For q = 3, our results confirm via theoretical arguments properties, deduced by computer searches performed by Ebert and Hirschfeld, for the largest partial spreads of H(3,9). An overview of the status regarding these results is given in two summarizing tables. The similar results for the classical symplectic and orthogonal polar spaces are presented in De Beule et al. [8].      

Some New Upper Bounds for the Size of Partial Ovoids in Slim Generalized Polygons and Generalized Hexagon of Order (s, s3)

From an elementary observation, we derive some upper bounds for the number of mutually opposite points in the classical generalized polygons having 3 points on each line. In particular, it follows that the Ree-Tits generalized octagon O(2) of order (2, 4) has no ovoids. Also, we deduce from another observation a similar upper bound in any generalized hexagon of order (s, s ³).     

Translation Ovoids and Spreads of the Generalized Hexagon H(q)

Ovoids of the finite classical generalized hexagon H(q) that are translation with respect to a point are classified. By duality, translation spreads with respect to a line are classified when the characteristic is three. When the characteristic is not equal to three, it is shown that there are no ovoids that are translation with respect to a flag.     

Translation Ovoids of Generalized Quadrangles and Hexagnos

We define the notion of a translation ovoid in the classical generalized quadrangles and hexagons of order q, and we enumerate all known examples; translation spreads are defined dually. A modification of the known ovoids in the generalized hexagon H(q), q=3^2h+1, yields new ovoids of that hexagon. Dualizing and projecting along reguli, we obtain an alternative construction of the Roman ovoids due to Thas and Payne. Also, we construct a new translation spread in H(q) for any 1 mod 3, q odd, with the property that any projection along reguli yields the classical ovoid in the generalized quadrangle Q(4,q). Finally, we prove that for q odd, the new example is the only non-Hermitian translation spread in H(q) with the property that any projection along reguli yields the classical ovoid in Q(4,q).     

On the Sporadic Semifield Flock

We obtain the BLT-set associated with the sporadic semifield flock of the quadratic cone in PG(3,3⁵) as the complete intersection of the Payne–Thas and the Kantor–Knuth ovoids of the parabolic quadric Q(4,3⁵. Also, we give an alternative construction of the Penttila–Williams ovoid of Q(4,3⁵).     

Blocking sets and semifields

We find a relationship between semifield spreads of PG(3,q), small Rédei minimal blocking sets of PG(2,q²), disjoint from a Baer subline of a Rédei line, and translation ovoids of the hermitian surface H(3,q²).     

Partial Ovoids in Classical Finite Polar Spaces

Ovoids in finite polar spaces are related to many other objects in finite geometries. In this article, we prove some new upper bounds for the size of partial ovoids in Q ^–(2n+1,q) and W(2n+ 1,q). Further, we give a combinatorial proof for the non-existence of ovoids of H(2n +1,q ²) for n>q ³.     

Antidesigns and regularity of partial spreads in dual polar graphs

We give several examples of designs and antidesigns in classical finite polar spaces. These types of subsets of maximal totally isotropic subspaces generalize the dualization of the concepts of m ‐ovoids and tight sets of points in generalized quadrangles. We also consider regularity of partial spreads and spreads. The techniques that we apply were developed by Delsarte. In some polar spaces of small rank, some of these subsets turn out to be completely regular codes. © 2010 Wiley Periodicals, Inc. J Combin Designs 19: 202‐216, 2011     

Intersections of Hermitian and Ree ovoids in the generalized hexagon H(q)

For q = 3^2h+1, h ≥ 0, we investigate the intersections of Hermitian and Ree ovoids of the generalized hexagon H(q). © 1996 John Wiley & Sons, Inc.     

Some p-Ranks Related to Orthogonal Spaces

We determine the p-rank of the incidence matrix of hyperplanes of PG(n, p ^e) and points of a nondegenerate quadric. This yields new bounds for ovoids and the size of caps in finite orthogonal spaces. In particular, we show the nonexistence of ovoids in and . We also give slightly weaker bounds for more general finite classical polar spaces. Another application is the determination of certain explicit bases for the code of PG(2, p) using secants, or tangents and passants, of a nondegenerate conic.     

Overgroups of Cyclic Sylow Subgroups of Linear Groups

We use a theorem of Guralnick, Penttila, Praeger, and Saxl to classify the subgroups of the general linear group (of a finite dimensional vector space over a finite field) which are overgroups of a cyclic Sylow subgroup. In particular, our results provide the starting point for the classification of transitive m-systems; which include the transitive ovoids and spreads of finite polar spaces. We also use our results to prove a conjecture of Cameron and Liebler on irreducible collineation groups having equally many orbits on points and on lines.     

New families of ovoids in O + 8

We construct four new infinite families of ovoids in the 8-dimensional orthogonal geometry O _inf8 ^sup+ . We determine the automorphism groups of these ovoids and we show that the two sporadic ovoids recently found by Cooperstein [2] and Shult [11] are members of our families.     

Applications of Number Theory to Ovoids and Translation Planes

In this paper we show that for each prime p7 there exists a translation plane of order p ² of Mason–Ostrom type. These planes occur as six-dimensional ovoids being projections of the eight-dimensional binary ovoids of Conway, Kleidman and Wilson. In order to verify the existence of such projections we prove certain properties of two particular quadratic forms using classical methods form number theory.     

Desarguesian and Unitary complete partial ovoids

The Desarguesian and Unitary ovoids of $\mathcal{Q}^{+}(7,q)$ are contained in a subvariety of the Grassmannian. We generalize this result, obtaining two families of complete partial ovoids of polar spaces: the first, with stabilizer PGL?(2,q ^t ) and including the family constructed by A. Cossidente and O.H. King for even q, and the second, with stabilizers contained in PGU?(r,q).     

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 q ^{2n ? 1} ? 1 MAXMOLS(q ²ⁿ ), for n ≥ 2 and q a power of two, and for n = 2 and q a power of three.     

The classical 1-system of Q?(7, q) and two-character sets

We will show that associated with the classical 1-system of the elliptic quadric Q ⁻(7, q) are certain infinite families of two-character sets with respect to hyperplanes, and partial ovoids of Q ⁺(15, q).