In this article, we examine sets of lines in PG(d, F) meeting each hyperplane in a generator set of points. We prove that such a set has to contain at least 1.5d lines if the field F has more than 1.5d elements, and at least 2d − 1 lines if the field F is algebraically closed. We show that suitable 2d − 1 lines constitute such a set (if |F| ≥ 2d − 1), proving that the lower bound is tight over algebraically closed fields. At last, we will see that the strong (s, A) subspace designs constructed by Guruswami and Kopparty [3] have better (smaller) parameter A than one would think at first sight. * -fold blocking sets are the same, since two distinct points span the line connecting these points.Definition 1 (Multiple blocking set). A set B of points in the projective space P is a t-fold blocking set with respect to hyperplanes, if each hyperplane Π ⊂ P meets B in at least t points. One can define t-fold blocking sets with respect to lines, planes, etc. similarly.The definition of the t-fold blocking set does not say anything more about the intersections with hyperplanes. In a projective space of dimension d ≥ 3, a d-fold blocking set can intersect a hyperplane Π in such a set of d points which is contained in a proper subspace of Π. Thus (in higher dimensions), a natural specialization of multiple blocking sets would be the following. (Since in higher dimension a projective space is always over a field, we use the special notation PG(d, F) instead of the general P.)Definition 2 (Generator set). A set G of points in the projective space PG(d, F) is a generator set with respect to hyperplanes, if each hyperplane Π ⊂ PG(d, F) meets G in a 'generator system' of Π, that is, G ∩ Π spans Π, in other words this intersection is not contained in any hyperplane of Π.(Hyperplanes of hyperplanes are subspaces in PG(d, F) of co-dimension two.)Example 3. In a projective plane PG(2, q 2 ) there exist two disjoint Baersubgeometries. These together constitute a 2-fold blocking set, and thus, a generator set consisting of 2q 2 + 2q + 2 points.Remark 4. In PG(d, q d ), d disjoint subgeometries of order q together constitute a d-fold blocking set. But it is not obvious whether this example is only a d-fold blocking set or it could be also a generator set (if we choose the subgeometries in a proper way).Héger and Takáts had the idea to search for generator set which is the union of some disjoint lines and Patkós gave an example for such a 'determining set' as the union of the points of 2d + 2 distinct lines, using probabilistic method. They gave the name 'higgledy-piggledy' to the property of such sets of lines. We investigate their idea. Hyperplane-generating sets of linesThe trivial examples for multiple blocking sets are the sets of disjoint lines: If B is the set of points of t disjoint lines then B is a t-fold blocking set (with respect to hyperplanes). Héger, Patkós and Takáts [1] suggested to search generator sets in such a form. (Though there can exist smaller examples.)

We study codewords of small weight in the codes arising from Desarguesian projective planes. We first of all improve the results of K. Chouinard on codewords of small weight in the codes arising from PG(2, p), p prime. Chouinard characterized all the codewords up to weight 2 p in these codes. Using a particular basis for this code, described by Moorhouse, we characterize all the codewords of weight up to 2 p + ( p − 1)/2 if p ≥ 11. We then study the codes arising from PG(2, q = q 3 0 ). In particular, for q 0 = p prime, p ≥ 7, we prove that the codes have no codewords with weight in the interval [q + 2, 2q − 1]. Finally, for the codes of PG(2, q), q = p h , p prime, h ≥ 4, we present a discrete spectrum for the 123 26 V. Fack et al.weights of codewords with weights in the interval [q + 2, 2q − 1]. In particular, we exclude all weights in the interval [3q/2, 2q − 1].

In this article we prove a theorem about the number of directions determined by less then q affine points, similar to the result of Blokhuis et. al.[3] on the number of directions determined by q affine points.

In this article, we investigate collections of 'well-spread-out' projective (and linear) subspaces. Projective k-subspaces in PG(d, F) are in 'higgledy-piggledy arrangement' if they meet each projective subspace of co-dimension k in a generator set of points. We prove that the set H of higgledy-piggledy k-subspaces has to contain more than min |F|, k i=0 ⌊ d−k+i i+1 ⌋ elements. We also prove that H has to contain more than (k + 1) · (d − k) elements if the field F is algebraically closed. * −r−h(s−1) for the first Guruswami-Kopparty construction, and A ≤ (m−s)s m−r−h(s−1) for the second Guruswami-Kopparty construction.