Péter Sziklai

Linear Point Sets and Rédei Type k-blocking Sets in PG(n, q)

In this paper, k-blocking sets in PG(n, q), being of Rédei type, are investigated. A standard method to construct Rédei type k-blocking sets in PG(n, q) is to construct a cone having as base a Rédei type k′-blocking set in a subspace of PG(n, q). But also other Rédei type k-blocking sets in PG(n, q), which are not cones, exist. We give in this article a condition on the parameters of a Rédei type k-blocking set of PG(n, q = ph), p a prime power, which guarantees that the Rédei type k-blocking set is a cone. This condition is sharp. We also show that small Rédei type k-blocking sets are linear.

Covers and blocking sets of classical generalized quadrangles

This article discusses two problems on classical generalized quadrangles. It is known that the generalized quadrangle Q(4,q) arising from the parabolic quadric in PG(4,q) has a spread if and only if q is even. Hence, for q odd, the problem arises of the cardinality of the smallest set of lines of Q(4,q) covering all points of Q(4,q). We show in this paper that this set of lines must contain more than q2+1+(q−1)/3 lines. We also show that Q(4,q), q even, does not contain minimal covers of sizes q2+1+r when q⩾32 and 0<r⩽q. To obtain this latter result, we generalize a result on minimal covers of lines in PG(3,q) to minimal covers of lines of the classical generalized quadrangles. This result is then also used to study minimal blocking sets of the non-singular generalized quadrangle U(4,q2) arising from the Hermitian variety in PG(4,q2). It is known that U(4,q2) does not have an ovoid. Here, we show that it also does not contain minimal blocking sets of sizes q5+2, q5+3 and q5+4 except maybe for small values of q.

A bound on the number of points of a plane curve

A conjecture is formulated for an upper bound on the number of points in PG(2,q) of a plane curve without linear components, defined over GF(q). We prove a new bound which is half-way from the known bound to the conjectured one. The conjecture is true for curves of low or high degree, or with rational singularity.

On the Spectrum of the Sizes of Maximal Partial Line Spreads in PG(2n,q), n 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. Szőnyi, 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 q2n-1+q2n-3+...+q3+1. The largest size for the maximal partial line spreads constructed in [A. Gács and T. Szőnyi, 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.

Two Remarks on Blocking Sets and Nuclei in Planes of Prime Order

In this paper we characterize a sporadic non-Rédei Type blocking set of PG(2,7) having minimum cardinality, and derive an upper bound for the number of nuclei of sets in PG(2,q) having less than q+1 points. Our methods involve polynomials over finite fields, and work mainly for planes of prime order.

Minimal Covers of the Klein Quadric

A t-cover of a quadric Q is a set C of t-dimensional subspaces contained in Q such that every point of Q belongs to at least one element of C. We consider t-covers of the Klein quadric Q+(5, q). For t=2, we show that a 2-cover has at least q2+q elements, and we give an exact description of the examples of this cardinality. For t=1, we show that a 1-cover has at least q3+2q+1 elements, and we give examples of covers of that size.

Reconstruction of matrices from submatrices

For an arbitrary matrix A of n x n symbols, consider its submatrices of size k x k, obtained by deleting n―k rows and n―k columns. Optionally, the deleted rows and columns can be selected symmetrically or independently. We consider the problem of whether these multisets determine matrix A. Following the ideas of Krasikov and Roditty in the reconstruction of sequences from subsequences, we replace the multiset by the sum of submatrices. For k > cn 2/3 we prove that the matrix A is determined by the sum of the k x k submatrices, both in the symmetric and in the nonsymmetric cases.

Partial flocks of the quadratic cone

We prove that in PG(3, q), q > 19, a partial flock of a quadratic cone with q - e planes, can be extended to a unique flock if e < 1/4 √q.

Nuclei of pointsets in PG (n,q)

Abstract In this paper we generalize the definition of nucleus. Using polynomials we get very similar but more general results involving the number of nuclei as in Blokhuis [1, 2] or Gacs et al. [4].

An extension of the direction problem

Abstract Let U be a point set in the n -dimensional affine space AG ( n , q ) over the finite field of q elements and 0 ≤ k ≤ n − 2 . In this paper we extend the definition of directions determined by U : a k -dimensional subspace S k at infinity is determined by U if there is an affine ( k + 1 ) -dimensional subspace T k + 1 through S k such that U ∩ T k + 1 spans T k + 1 . We examine the extremal case | U | = q n − 1 , and classify point sets not determining every k -subspace in certain cases.