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 and < N, a {v _{+ 1}, v ; N, q}-minihyper consists of a sum of -spaces, we show that the excess points of an s-cover with excess of PG(N, q), (s + 1)|(N + 1), form a sum of s-spaces, and that no maximal partial s-spreads with deficiency 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].     

Multiple Blocking Sets in PG(n, q), n > 3

This article discusses minimal s-fold blocking sets B in PG (n, q), q = p^h, p prime, q > 661, n > 3, of size |B| > sq + c ^p q ^2/3 - (s - 1) (s - 2)/2 (s > min (c ^p q ^1/6, q ^1/4/2)). It is shown that these s-fold blocking sets contain the disjoint union of a collection of s lines and/or Baer subplanes. To obtain these results, we extend results of Blokhuis–Storme–Szönyi on s-fold blocking sets in PG(2, q) to s-fold blocking sets having points to which a multiplicity is given. Then the results in PG(n, q), n 3, are obtained using projection arguments. The results of this article also improve results of Hamada and Helleseth on codes meeting the Griesmer bound.     

Scattered Spaces with Respect to a Spread in PG(n,q)

A scattered subspace of PG(n-1,q) with respect to a (t-1)-spread S is a subspace intersecting every spread element in at most a point. Upper and lower bounds for the dimension of a maximum scattered space are given. In the case of a normal spread new classes of two intersection sets with respect to hyperplanes in a projective space are obtained using scattered spaces.     

Results on Maximal Partial Spreads in PG(3, p 3) and on Related Minihypers

This article classifies all {(q + 1), 3, q}-minihypers, small, q = p ^h ₀, h 1, for a prime number p ₀ 7, which arise from a maximal partial spread of deficiency . When q is a third power, the minihyper is the disjoint union of projected PG(5, )'s; when q is a square, also Baer subgeometries PG(3, ) can occur. This leads to a discrete spectrum for the small values of the deficiency of the corresponding maximal partial spreads.     

Normal Spreads

In Dedicata 16 (1984), pp. 291–313, the representation of Desarguesian spreads of the projective space PG(2t – 1, q) into the Grassmannian of the subspaces of rank t of PG(2t – 1, q) has been studied. Using a similar idea, we prove here that a normal spread of PG(rt – 1,q) is represented on the Grassmannian of the subspaces of rank t of PG(rt – 1, q) by a cap V _{r, t} of PG(r ^t – 1, q), which is the GF(q)-scroll of a Segre variety product of t projective spaces of dimension (r – 1), and that the collineation group of PG(r ^t – 1, q) stabilizing V _{r, t} acts 2-transitively on V _{r, t} . In particular, we prove that V _{3, 2} is the union of q ² – q + 1 disjoint Veronese surfaces, and that a Hermitan curve of PG(2, q ²) is represented by a hyperplane section U of V _{3, 2}. For q 0,2 (mod 3) the algebraic variety U is the unitary ovoid of the hyperbolic quadric Q ⁺ (7, q) constructed by W. M. Kantor in Canad. J. Math., 5 (1982), 1195–1207. Finally we study a class of blocking sets, called linear, proving that many of the known examples of blocking sets are of this type, and we construct an example in PG(3, q ²). Moreover, a new example of minimal blocking set of the Desarguesian projective plane, which is linear, has been constructed by P. Polito and O. Polverino.     

A Borsuk–Ulam Theorem for Maps from a Sphere to a Generalized Manifold

H. J. Munkholm obtained a generalization for topological manifolds of the famous Borsuk–Ulam type theorem proved by Conner and Floyd. The purpose of this paper is to prove a version of Conner and Floyd's theorem for generalized manifolds.     

A Bose-Burton Theorem for Elliptic Polar Spaces

In bose&burton, Bose and Burton determined the smallest point sets of PG(d, q) that meet every subspace of PG(d, q) of a given dimension c. In this paper an equivalent result for quadrics of elliptic type is obtained. It states the folloing. For 0 c n - 1 the smallest point set of the elliptic quadric Q ^-(2n + 1, q) that meets every singular subspace of dimension c of Q ^-(2n + 1, q) has cardinality (q ⁿ⁺¹ + q ^c )(q ^n-c - 1)/(q - 1). Furthermore, the point sets of the smallest cardinality are classified.     

Toward the classification of biangular harmonic frames

Equiangular tight frames (ETFs) and biangular tight frames (BTFs) – sets of unit vectors with basis-like properties whose pairwise absolute inner products admit exactly one or two values, respectively – are useful for many applications. A well-understood class of ETFs are those which manifest as harmonic frames – vector sets defined in terms of the characters of finite abelian groups – because they are characterized by combinatorial objects called difference sets.This work is dedicated to the study of the underlying combinatorial structures of harmonic BTFs. We show that if a harmonic frame is generated by a divisible difference set, a partial difference set or by a special structure with certain Gauss summing properties – all three of which are generalizations of difference sets that fall under the umbrella term “bidifference set” – then it is either a BTF or an ETF. However, we also show that the relationship between harmonic BTFs and bidifference sets is not as straightforward as the correspondence between harmonic ETFs and difference sets, as there are examples of bidifference sets that do not generate harmonic BTFs. In addition, we study another class of combinatorial structures, the nested divisible difference sets, which yields an example of a harmonic BTF that is not generated by a bidifference set.     

Blocking Sets in André Planes

We construct new examples of blocking sets in some non-Desarguesian translation planes.     

On the Spectrum of the Sizes of Maximal Partial Line Spreads in <Emphasis Type="Italic">PG</Emphasis>(2<Emphasis Type="Italic">n</Emphasis>,<Emphasis Type="Italic">q</Emphasis>), <Emphasis Type="Italic">n</Emphasis> ≥ 3

A lot of research has been done on the spectrum of the sizes of maximal partial spreads in PG(3,q) [P. Govaerts and L. Storme, Designs Codes and Cryptography, Vol. 28 (2003) pp. 51–63; O. Heden, Discrete Mathematics, Vol. 120 (1993) pp. 75–91; O. Heden, Discrete Mathematics, Vol. 142 (1995) pp. 97–106; O. Heden, Discrete Mathematics, Vol. 243 (2002) pp. 135–150]. In [A. Gács and T. Sznyi, Designs Codes and Cryptography, Vol. 29 (2003) pp. 123–129], results on the spectrum of the sizes of maximal partial line spreads in PG(N,q), N 5, are given. In PG(2n,q), n 3, the largest possible size for a partial line spread is q^2n-1+q^2n-3+...+q³+1. The largest size for the maximal partial line spreads constructed in [A. Gács and T. Sznyi, Designs Codes and Cryptography, Vol. 29 (2003) pp. 123–129] is (q²ⁿ⁺¹–q)/(q²–1)–q³+q²–2q+2. This shows that there is a non-empty interval of values of k for which it is still not known whether there exists a maximal partial line spread of size k in PG(2n,q). We now show that there indeed exists a maximal partial line spread of size k for every value of k in that interval when q 9.J. Eisfeld: Supported by the FWO Research Network WO.011.96NP. Sziklai: The research of this author was partially supported by OTKA D32817, F030737, F043772, FKFP 0063/2001 and Magyary Zoltan grants. The third author is grateful for the hospitality of Ghent University.     

Chern Classes of Gauss–Manin Bundles of Weight 1 Vanish

We show that the Chern character of a variation of polarized Hodge structures of weight one with nilpotent residues at dies up to torsion in the Chow ring, except in codimension 0.     

A Remark on an Algebraic Riemann—Roch Formula for Flat Bundles

Let f:X S be a smooth projective morphism over an algebraically closed field, with X and S regular. When E, ) is a flat bundle over X, then its Gauss–Manin bundles on S have a flat connection and one may ask for a Riemann–Roch formula relating the algebraic Chern–Simons and Cheeger–Simons invariants. We give an answer for X = Y × S, f = projection. The method of proof is inspired by the work of Hitchin and Simpson.     

Blocking Sets in (υ,{2, 4}, 1)-Designs

The existence of blocking sets in (, {2, 4}, 1)-designs is examined. We show that for 0, 3, 5, 6, 7, 8, 9, 11 (mod 12>), blocking sets cannot exist. We prove that for each 1, 2, 4 (mod 12) there is a (, {2, 4}, 1)-design with a blocking set with three possible exceptions. The case 10 (mod 12) is still open; we consider the first four values of in this situation.     

A few more large sets of t-designs

We construct several new large sets of t-designs that are invariant under Frobenius groups, and discuss their consequences. These large sets give rise to further new large sets by means of known recursive constructions including an infinite family of large sets of 3 − (v, 4, λ) designs. © 1998 John Wiley & Sons, Inc. J Combin Designs 6: 293–308, 1998