Projective Geometries Over Finite Fields

This book is an account of the combinatorics of projective spaces over a finite field, with special emphasis on one and two dimensions. With its successor volumes, Finite projective spaces over three dimensions (1985), which is devoted to three dimensions, and General Galois geometries (1991), on a general dimension, it provides the only comprehensive treatise on this area of mathematics. The area is interesting in itself, but is important for its applications to coding theory and statistics, and its use of group theory, algebraic geometry, and number theory. This new edition is a complete reworking, containing extensive revisions, particularly in the chapters on generalities, the geometry of arcs in ovals, the geometry of arcs of higher degree, and blocking sets. Part I gives a survey of finite fields and an outline of the fundamental properties of projective spaces and their automorphisms; it includes the properties of algebraic varieties and curves used throughout the book and in the companion volumes. Part II covers, in an arbitrary dimension, the properties of subspaces, of partitions into both subspaces and subgeometries, and of quadrics and Hermitian varieties, as well as polarities. Part III is a detailed account of the line and plane; with little reference to the generalities from Parts I and II, the author revisits fundamental properties of the plane and then describes the structure of arcs and their relation to curves. This part includes chapters on blocking sets and on small planes (those with orders up to thirteen). With a comprehensive bibliography containing over 3,000 items, this volume will prove invaluable to researchers in finite geometry, coding theory and combinatorics.