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 a comprehensive treatise of 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 edition is a
complete reworking of the first edition. The chapters bear almost
the same titles as the first edition, but every chapter has been
changed. The most significant changes are to Chapters 2, 10, 12,
13, which respectively describe generalities, the geometry of arcs
in ovals, the geometry of arcs of higher degree, and blocking sets.
The book is divided into three parts. The first part comprises two
chapters, the first of which is a survey of finite fields; the
second outlines the fundamental properties of projective spaces and
their automorphisms, as well as properties of algebraic varieties
and curves, in particular, that are used in the rest of the book
and the accompanying two volumes. Parts II and III are entirely
self-contained; all proofs of results are given. The second part
comprises Chapters 3 to 5. They cover, in an arbitrary dimension,
the properties of subspaces such as their number and
characterization, 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 the
plane. In the plane, fundamental properties are first revisited
without much resort to the generalities of Parts I and II. Then,
the structure of arcs and their relation to curves is described;
this includes arcs both of degree two and higher degrees. There are
further chapters on blocking sets and on small planes, which means
of orders up to thirteen. A comprehensive bibliography of more than
3000 items is provided. At the end of each chapter is a section,
Notes and References, which attributes proofs, includes further
comments, and lists every relevant reference from the bibliography.
General
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