Maska Law

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,q^2), 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,q^2), 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 m -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,q2).

Flag-transitive symmetric 2-(96,20,4)-designs

We construct four flag-transitive symmetric designs having 96 points, blocks of size 20, and 4 blocks on each point-pair. Moreover we prove that these are the only such designs. Our classification completes the classification of flag-transitive, point-imprimitive, symmetric designs with the (constant) number of blocks on a point-pair at most four.

Finite Line-transitive Linear Spaces： Theory and Search Strategies

In 1991, Weidong Fang and Huiling Li proved that there are only finitely many non-trivial linear spaces that admit a line-transitive, point imprimitive group action, for a given value of gcd(k, r), where k is the line size and r is the number of lines on a point. The aim of this paper is to make that result effective. We obtain a classification of all linear spaces with this property having gcd(k, r) ≤ 8. To achieve this we collect together existing theory, and prove additional theoretical restrictions of both a combinatorial and group theoretic nature. These are organised into a series of algorithms that, for gcd(k, r) up to a given maximum value, return a list of candidate parameter values and candidate groups. We examine in detail each of the possibilities returned by these algorithms for gcd(k, r) ≤ 8, and complete the classification in this case. 2000 Mathematics Subject Classification: 05B05, 05B25, 20B25

Construction of BLT-sets over small fields

The computer algebra system MAGMA is used to search for BLT-sets of the nonsingular parabolic quadric Q(4, q). In total, 28 new BLT-sets for 27 ≤ q ≤ 125 are presented, these giving rise to 158 new flocks of the quadratic cone in PG(3, q).

Symmetries of BLT-Sets

We do the tentative beginnings of a study of BLT-sets of generalised quadrangles via their symmetries. In particular, the study of whorls about a line leads us to hyperbolic reflections preserving a BLT-set of Q(4, q).

Flocks and Partial Flocks of Hyperbolic Quadrics via Root Systems

We construct three infinite families of partial flocks of sizes 12, 24 and 60 of the hyperbolic quadric of PG(3, q), for q congruent to -1 modulo 12, 24, 60 respectively, from the root systems of type D4, F4, H4, respectively. The smallest member of each of these families is an exceptional flock. We then characterise these partial flocks in terms of the rectangle condition of Benz and by not being subflocks of linear flocks or of Thas flocks. We also give an alternative characterisation in terms of admitting a regular group fixing all the lines of one of the reguli of the hyperbolic quadric.

Some Flocks in Characteristic 3

A family of flocks is presented in characteristic 3. From the standard theory, there are associated generalised quadrangles, translation planes, and BLT-sets, leading to further flocks.