Lines in Higgledy-Piggledy Arrangement

Abstract

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.)

On resolving sets in the point-line incidence graph of PG(n, q)

“…There are two natural generalizations of this planar result in higher dimensional spaces: one can consider either the point-hyperplane incidence graph, or the point-line incidence graph of PG(n, q). In the former case resolving sets are connected with lines in a higgledypiggledy arrangement which were investigated by Fancsali and Sziklai [9]. Their results were recently improved by the authors of this paper [5].…”

“…Resolving sets for incidence graphs of some linear spaces were investigated by several authors [4,9,10,12]. In these cases much better bounds than the general ones are known.…”

Small Strong Blocking Sets by Concatenation

“…Strong blocking sets were introduced in [16]. In [19], strong blocking sets are referred to as generator sets and they are constructed as union of disjoint lines. In [10] they were reintroduced, with the name of cutting blocking sets, in order to construct a particular family of minimal codes.…”

“…A point set B is called a blocking set if any hyperplane of PG(N, q) contains at least one point of B. A blocking set B is strong if any hyperplane section of B generates the hyperplane itself; see [10,16,19]. Although blocking sets were deeply investigated in the last decades (see for example [9] and references therein), much less is known about strong blocking sets.…”

“…It is worth mentioning that minimal codes form a class of asymptotically good codes [1,14]. Families of short minimal codes (via their equivalence with strong blocking sets) were provided in [5,21] also in connection with line spreads or sets of lines in higgledy-piggledy arrangement [19]. Apart from a few small dimensional cases, none of these constructions were effective.…”