J. W. P. Hirschfeld

General Galois geometries

This book is the second edition of the third and last volume of a treatise on projective spaces over a finite field, also known as Galois geometries. This volume completes the trilogy comprised of plane case (first volume) and three dimensions (second volume). This revised edition includes much updating and new material. It is a mostly self-contained study of classical varieties over a finite field, related incidence structures and particular point sets in finite n-dimensional projective spaces. General Galois Geometries is suitable for PhD students and researchers in combinatorics and geometry. The separate chapters can be used for courses at postgraduate level.

The packing problem in statistics, coding theory and finite projective spaces

In the last few decades, nite projective spaces or, equivalently, Galois geometries have been studied intensively. Apart from being an interesting and exciting area in combinatorics with beautiful results, this eld has many connections with statistics and coding theory. Indeed, several problems have equivalent formulations in the di erent areas. A problem, rst studied in statistics by Fisher [56, 57], has proved to be equivalent to a problem in geometry [25]. In [22, 24], Bose generalized this application of nite projective geometry for the design of experiments and called it the packing problem. He also presented, in 1961, connections between the design of experiments and coding theory [24, 25]. The central problem posed in these articles is the determination of m(n; r; s;N; q), the largest size of a point set, as de ned in Section 1.2. After initial consideration by Bose and his followers as a statistical problem, and further interest by Kustaanheimo [94] and other Finnish astronomers, the topic was taken up by Segre [128, 132] and his followers, perhaps because of Segre’s interest in algebraic geometry over nite elds. Using geometric methods, many fundamental results were obtained. Coding theory provides a second motivation for these problems, which have equivalent formulations in nite projective spaces and coding theory. This amounts in coding theory to studying the row space of a generator matrix of a code and in Galois geometry to studying the column space. The classical example, that is, the equivalence of linear maximum distance separable (MDS) codes and arcs in projective spaces, has

The packing problem in statistics, coding theory and finite projective spaces: Update 2001

This article updates the authors’ 1998 survey [134] on the same theme that was written for the Bose Memorial Conference (Colorado, June 7–11, 1995). That article contained the principal results on the packing problem, up to 1995. Since then, considerable progress has been made on different kinds of subconfigurations.

Complete Arcs in Planes of Square Order

Large arcs in cyclic planes of square order are constructed as orbits of a subgroup of a group whose generator acts as a single cycle. In the Desarguesian plane of even square order, this gives an example of an are achieving the upper bound for complete arcs other than ovals.

The weight hierarchy of higher dimensional Hermitian codes

Studies a class of projective systems and linear codes corresponding to Hermitian varieties over finite fields. The weight hierarchy, also known as the set of generalized Hamming weights, of the code is calculated. The higher weight distribution is also found. >

Complete Systems of Lines on a Hermitian Surface over aFinite Field

The aim is to find the maximum size of a set of mutually ske lines on a nonsingular Hermitian surface in PG(3, q) for various values of q. For q = 9 such extremal sets are intricate combinatorial structures intimately connected ith hemisystems, subreguli, and commuting null polarities. It turns out they are also closely related to the classical quartic surface of Kummer. Some bounds and examples are also given in the general case.

Ovoids, spreads and m-systems of finite classical polar spaces

In this chapter, ovoids, spreads and m-systems of finite classical polar spaces are introduced. Also SPG-reguli, SPG-systems, BLT-sets and sets with the BLT-property are defined. The main results on these topics are given, all without proof.

Intersections in Projective Space II

A complete classification is given of pencils of quadrics in projective space of three dimensions over a finite field, where each pencil contains at least one non-singular quadric and where the base curve is not absolutely irreducible. This leads to interesting configurations in the space such as partitions by elliptic quadrics and by lines.

On Plane Maximal Curves

AbstractThe number N of rational points on an algebraic curve of genus g over a finite field