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     

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).     

A spectrum result on minimal blocking sets with respect to the planes of PG(3, q), q odd

This article presents a spectrum result on minimal blocking sets with respect to the planes of PG(3, q), q odd. We prove that for every integer k in an interval of, roughly, size [q ²/4, 3q ²/4], there exists such a minimal blocking set of size k in PG(3, q), q odd. A similar result on the spectrum of minimal blocking sets with respect to the planes of PG(3, q), q even, was presented in Rößing and Storme (Eur J Combin 31:349–361, 2010). Since minimal blocking sets with respect to the planes in PG(3, q) are tangency sets, they define maximal partial 1-systems on the Klein quadric Q ⁺(5, q), so we get the same spectrum result for maximal partial 1-systems of lines on the Klein quadric Q ⁺(5, q), q odd.     

Regular graphs constructed from the classical generalized quadrangle Q(4, q)

Let c_k be the smallest number of vertices in a regular graph with valency k and girth 8. It is known that c_{k + 1}?2(1 + k + k² + k³) with equality if and only if there exists a finite generalized quadrangle of order k. No such quadrangle is known when k is not a prime power. In this case, small regular graphs of valency k + 1 and girth 8 can be constructed from known generalized quadrangles of order q>k by removing a part of its structure. We investigate the case when q = k + 1 is a prime power, and try to determine the smallest graph under consideration that can be constructed from a generalized quadrangle of order q. This problem appears to be much more difficult than expected. We have general bounds and improve these for the classical generalized quadrangle Q(4, q), q even. © 2010 Wiley Periodicals, Inc. J Combin Designs 19:70‐83, 2010     

Ovoidal blocking sets and maximal partial ovoids of Hermitian varieties

In Mazzocca et al. (Des. Codes Cryptogr. 44:97–113, 2007), large minimal blocking sets in PG(3, q ²) and PG(4, q ²) have been constructed starting from ovoids of PG(3, q), Q(4, q) and Q(6, q). Some of these can be embedded in a Hermitian variety as maximal partial ovoids. In this paper, the geometric conditions assuring these embeddings are established.     

Generalized quadrangles of order (q,q 2),q even,containingW(q) as a subquadrangle

In this paper we give a characterization of the generalized quadrangleQ(5,q),q even, in terms of ovoids of its subquadrangles.     

Commuting polarities and maximal partial ovoids of H(4,q2)

In PG(4,q²), q odd, let Q(4,q²) be a non‐singular quadric commuting with a non‐singular Hermitian variety H(4,q²). Then these varieties intersect in the set of points covered by the extended generators of a non‐singular quadric Q₀ in a Baer subgeometry Σ₀ of PG(4,q²). It is proved that any maximal partial ovoid of H(4,q²) intersecting Q₀ in an ovoid has size at least 2(q²+1). Further, given an ovoid O of Q₀, we construct maximal partial ovoids of H(4,q²) of size q³+1 whose set of points lies on the hyperbolic lines 〈P,X〉 where P is a fixed point of O and X varies in O\{P}. © 2009 Wiley Periodicals, Inc. J Combin Designs 17: 307–313, 2009     

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.     

Tight sets and m-ovoids of finite polar spaces

An intriguing set of points of a generalised quadrangle was introduced in [J. Bamberg, M. Law, T. Penttila, Tight sets and m-ovoids of generalised quadrangles, Combinatorica, in press] as a unification of the pre-existing notions of tight set and m-ovoid. It was shown in [J. Bamberg, M. Law, T. Penttila, Tight sets and m-ovoids of generalised quadrangles, Combinatorica, in press] that every intriguing set of points in a finite generalised quadrangle is a tight set or an m-ovoid (for some m). Moreover, it was shown that an m-ovoid and an i-tight set of a common generalised quadrangle intersect in mi points. These results yielded new proofs of old results, and in this paper, we study the natural analogue of intriguing sets in finite polar spaces of higher rank. In particular, we use the techniques developed in this paper to give an alternative proof of a result of Thas [J.A. Thas, Ovoids and spreads of finite classical polar spaces, Geom. Dedicata 10 (1-4) (1981) 135-143] that there are no ovoids of H(2r,q²), Q⁻(2r+1,q), and W(2r−1,q) for r>2. We also strengthen a result of Drudge on the non-existence of tight sets in W(2r−1,q), H(2r+1,q²), and Q⁺(2r+1,q), and we give a new proof of a result of De Winter, Luyckx, and Thas [S. De Winter, J.A. Thas, SPG-reguli satisfying the polar property and a new semipartial geometry, Des. Codes Cryptogr. 32 (1-3) (2004) 153-166; D. Luyckx, m-Systems of finite classical polar spaces, PhD thesis, The University of Ghent, 2002] that an m-system of W(4m+3,q) or Q⁻(4m+3,q) is a pseudo-ovoid of the ambient projective space.     

Tight sets and <Emphasis Type="Italic">m</Emphasis>-ovoids of generalised quadrangles

The concept of a tight set of points of a generalised quadrangle was introduced by S. E. Payne in 1987, and that of an m-ovoid of a generalised quadrangle was introduced by J. A. Thas in 1989, and we unify these two concepts by defining intriguing sets of points. We prove that every intriguing set of points in a generalised quadrangle is an m-ovoid or a tight set, and we state an intersection result concerning these objects. In the classical generalised quadrangles, we construct new m-ovoids and tight sets. In particular, we construct m-ovoids of W(3,q), q odd, for all even m; we construct (q+1)/2-ovoids of W(3,q) for q odd; and we give a lower bound on m for m-ovoids of H(4,q ²).     

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).     

Elliptic ovoids and their rosettes in a classical generalized quadrangle of even order

Let Q₀ be the classical generalized quadrangle of order q = 2ⁿ(n≥2) arising from a non-degenerate quadratic form in a 5-dimensional vector space defined over a finite field of order q. We consider the rank two geometry \(\mathcal {X}\) having as points all the elliptic ovoids of Q₀ and as lines the maximal pencils of elliptic ovoids of Q₀ pairwise tangent at the same point. We first prove that \(\mathcal {X}\) is isomorphic to a 2-fold quotient of the affine generalized quadrangle Q?Q₀, where Q is the classical (q,q²)-generalized quadrangle admitting Q₀ as a hyperplane. Further, we classify the cliques in the collinearity graph Γ of \(\mathcal {X}\). We prove that any maximal clique in Γ is either a line of \(\mathcal {X}\) or it consists of 6 or 4 points of \(\mathcal {X}\) not contained in any line of \(\mathcal {X}\), accordingly as n is odd or even. We count the number of cliques of each type and show that those cliques which are not contained in lines of \(\mathcal {X}\) arise as subgeometries of Q defined over \(\mathbb {F}_{2}\).     

The uniqueness of 1-systems of W 5(q) satisfying the BLT-property, with q odd

In the symplectic polar space W ₅(q) every 1-system which satisfies the BLT-property (and then q is odd) defines a generalized quadrangle (GQ) of order (q ²,q ³). In this paper, we show that this 1-system is unique, so that the only GQ arising in this way is isomorphic to the classical GQ H(4,q ²), q odd.     

Notes on binary codes related to the O(5, q) generalized quadrangle for odd q

In this note we determine the dimensions of the binary codes spanned by the lines or by the point neighborhoods in the generalized quadrangle Sp(4, q) and its dual O(5, q), where q is odd. Several more general results are given. As a side result we find that if a square generalized quadrangle of odd order has an antiregular point, then all of its points are antiregular.On leave from the Indian Statistical Institute, Calcutta; research supported by a grant from NWO.     

On codewords in the dual code of classical generalised quadrangles and classical polar spaces

In [J.L. Kim, K. Mellinger, L. Storme, Small weight codewords in LDPC codes defined by (dual) classical generalised quadrangles, Des. Codes Cryptogr. 42 (1) (2007) 73-92], the codewords of small weight in the dual code of the code of points and lines of Q(4,q) are characterised. Inspired by this result, using geometrical arguments, we characterise the codewords of small weight in the dual code of the code of points and generators of Q⁺(5,q) and H(5,q²), and we present lower bounds on the weight of the codewords in the dual of the code of points and k-spaces of the classical polar spaces. Furthermore, we investigate the codewords with the largest weights in these codes, where for q even and k sufficiently small, we determine the maximum weight and characterise the codewords of maximum weight. Moreover, we show that there exists an interval such that for every even number w in this interval, there is a codeword in the dual code of Q⁺(5,q), q even, with weight w and we show that there is an empty interval in the weight distribution of the dual of the code of Q(4,q), q even. To prove this, we show that a blocking set of Q(4,q), q even, of size q²+1+r, where 0<r<(q+4)/6, contains an ovoid of Q(4,q), improving on [J. Eisfeld, L. Storme, T. Sz?nyi, P. Sziklai, Covers and blocking sets of classical generalised quadrangles, Discrete Math. 238 (2001) 35-51, Theorem 9].     

Subquadrangle m‐regular systems on generalized quadrangles

We introduce the notion of subquadrangle regular system of a generalized quadrangle. A subquadrangle regular system of order m on a generalized quadrangle of order (s, t) is a set ? of embedded subquadrangles with the property that every point lies on exactly m subquadrangles of ?. If m is one half of the total number of subquadrangles on a point, we call ? a subquadrangle hemisystem. We construct two infinite families of symplectic subquadrangle hemisystems of the Hermitian surface ??(3, q²), q odd, and two infinite families of symplectic subquadrangle hemisystems of ??₃(q²), q even. Some sporadic examples of symplectic subquadrangle regular systems of ??(3, q²) are also presented. © 2010 Wiley Periodicals, Inc. J Combin Designs 19:28‐41, 2010     

Ovoids and spreads of finite classical polar spaces

LetP be a finite classical polar space of rankr, r2. An ovoidO ofP is a pointset ofP, which has exactly one point in common with every totally isotropic subspace of rankr. It is proved that the polar spaceW _n (q) arising from a symplectic polarity ofPG(n, q), n odd andn > 3, that the polar spaceQ(2n, q) arising from a non-singular quadric inPG(2n, q), n > 2 andq even, that the polar space Q^–(2n + 1,q) arising from a non-singular elliptic quadric inPG(2n + 1,q), n > 1, and that the polar spaceH(n,q ²) arising from a non-singular Hermitian variety inPG(n, q ²)n even andn > 2, have no ovoids.LetS be a generalized hexagon of ordern (1). IfV is a pointset of order n³ + 1 ofS, such that every two points are at distance 6, thenV is called an ovoid ofS. IfH(q) is the classical generalized hexagon arising fromG ₂ (q), then it is proved thatH(q) has an ovoid iffQ(6, q) has an ovoid. There follows thatQ(6, q), q=3^2h+1, has an ovoid, and thatH(q), q even, has no ovoid.A regular system of orderm onH(3,q ²) is a subsetK of the lineset ofH(3,q ²), such that through every point ofH(3,q ²) there arem (> 0) lines ofK. B. Segre shows that, ifK exists, thenm=q + 1 or (q + l)/2.If m=(q + l)/2,K is called a hemisystem. The last part of the paper gives a very short proof of Segre's result. Finally it is shown how to construct the 4-(11, 5, 1) design out of the hemisystem with 56 lines (q=3).     

Characterizations of generalized quadrangles by generalized homologies

Let S = (P, B, I) be a generalized quadrangle of order (s, t). For x, y P, x y, let (x, y) be the group of all collineations of S fixing x and y linewise. If z {x, y}, then the set of all points incident with the line xz (resp. yz) is denoted by (resp. ). The generalized quadrangle S = (P, B, I) is said to be (x, y)-transitive, x y, if (x, y) is transitive on each set and . If S = (P, B, I) is a generalized quadrangle of order (s, t), s > 1 and t > 1, which is (x, y)-transitive for all x, y P with x y, then it is proved that we have one of the following: (i) S W(s), (ii) S Q(4, s), (iii) S H(4, s), (iv) S Q(5, s), (v) s = t² and all points are regular.     

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.     

Generalized quadrangles associated with G2(q)

If q ≡ 2 (mod 3), a generalized quadrangle with parameters q, q² is constructed from the generalized hexagon associated with the group G₂(q).