Summary We study the following two problems in finite projective, affine and polar spaces of dimension d. Given integers 0 ≤ s, t ≤ d what is the cardinality of the smallest set T consisting of subspaces of dimension t with the property that every subspace of dimension s is incident with at least one element of T? Likewise, what is the cardinality of the largest set T consisting of subspaces of dimension t with the property that every subspace of dimension s is incident with at most one element of T? Bose and Burton solved the case t = 0 of the first question for projective spaces in 1966. We survey the known results, present some open problems, and prove new results for finite polar spaces.

Introduction

Consider a finite projective, affine or polar space of dimension d and two integers s and t with 0 ≤ s, t ≤ d and the following problems.

Problem 1: What is the smallest cardinality of a set T of t-subspaces such that every s-subspace is incident with at least one element of T?

Problem 2: What is the largest cardinality of a set T of t-subspaces such that every s-subspace is incident with at most one element of T?

Much attention has been paid to particular cases of both problems in the last 30 years. This article gives a survey and several new results concerning Problem 1. Some results concerning Problem 2 are mentioned too.

Summary A method of using polynomials to describe objects in finite geometries is outlined and the problems where this method has led to a solution are surveyed. These problems concern nuclei, affine blocking sets, maximal arcs and unitals. In the case of nuclei these methods give lower bounds on the number of nuclei to a set of points in PG(n, q), usually dependent on some binomial coefficient not vanishing modulo the characteristic of the field. These lower bounds on nuclei lead directly to lower bounds on affine blocking sets with respect to lines. A short description of how linear polynomials can be used to construct maximal arcs in certain translation planes is included. A proof of the non-existence of maximal arcs in PG(2, q) when q is odd is outlined and some bounds are given as to when a (k, n)-arc can be extended to a maximal arc in PG(2, q). These methods can also be applied to unitals embedded in PG(2, q). One implication of this is that when q is the square of a prime a non-classical unital has a limited number of Baer sublines amongst its secants.

Introduction

The effectiveness of polynomials as a means of studying problems in finite geometries has become increasingly evident in the 1990's, although the first examples seem to date back to R. Jamison [38] in 1977 and A. E. Brouwer and A. A. Bruen and J. C. Fisher described the "Jamison method" as the following: reformulate the problem in terms of points of an affine space and associate suitable polynomials defined over the corresponding finite field; calculate. This is the approach employed in [19] too; in fact the main difference between [38] and [19] is that Jamison viewed the points of an affine space as elements of a finite field. In effect, this has the advantage of reducing the number of variables in the polynomials and allowing one to use simple arguments concerning the degree or the coefficients of a polynomial.

Summary Combinatorial designs have long had substantial application in the statistical design of experiments and in the theory of error-correcting codes. Applications in experimental and theoretical computer science have emerged, along with connections with the theory of cryptographic communication. This paper focuses on applications in the general area of communications, including cryptography and networking. Applications have been chosen to represent those in which design theory plays a useful, and sometimes central, role. Moreover, applications have been chosen to reflect in addition the genesis of new and interesting problems in design theory in order to treat the practical concerns. Of many candidates, thirteen applications areas have been included:

1. Optical orthogonal codes

2. Synchronous multiple access to channels

3. Group testing and superimposed codes

4. Erasure codes and information dispersal

5. Threshold and ramp schemes

6. Authentication codes

7. Resilient and correlation-immune functions

8. Multidrop networks

9. Channel graphs and interconnection networks

10. Partial match queries on files

11. Software testing

12. Disk layout and striping

13. (t, m, s)-nets and numerical integration

The theory of combinatorial designs continues to grow, in part as a consequence of the variety of these applications and the increasing depth of the connections with challenging problems on designs.

Summary The author rehearses his role in the development of the theory of matroids. The story starts in 1935 when he became an undergraduate at Trinity College, Cambridge, and started to collaborate with Leonard Brooks, Cedric Smith and Arthur Stone. It continues through his war-time work with codes and ciphers, followed by his return to Trinity in 1945, where his PhD thesis entitled “An Algebraic Theory of Graphs” foreshadowed his matroid papers published in 1958 and 1959. He describes the context in which he obtained the now well-known excluded minor conditions for a binary matroid to be regular and for a regular matroid to be graphic. He subsequently invented the whirl, and lectured on matroids at the 1964 Conference where the theory of matroids was first proclaimed to the world. This paper has two appendices: “Geometrical Terminology” and “Binary and Regular Matroids”.

As we all know, matroids made their appearance in the mathematical literature in 1935, in a paper of Hassler Whitney entitled “On the abstract properties of linear dependence” [17].

Whitney looked at a matrix and saw that some sets of columns were independent and some were not. There were even simple rules about this distinction. For example, any subset of an independent set of columns is independent—provided of course that you count the null set as independent. Also, if you had an independent set A you could make it into a bigger one by adding the right member of any independent set that was bigger than A.