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Zusatztext 'Snap it up !' Bulletin London mathematical Society Informationen zum Autor Dr J. W. P. Hirschfeld, School of Mathematical Sciences, University of Sussex, Brighton, BN1 9QH. Tel.: +44 1273 678080; fax: +44 1273 678097 Klappentext 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 isa 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 andtheir 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 Hermitianvarieties, 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 includeschapters 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 theoryand combinatorics. Zusammenfassung 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 p...
