Book•
Projective geometries over finite fields
24 Jan 1980-
TL;DR: The first properties of the plane can be found in this article, where the authors define the following properties: 1. Finite fields 2. Projective spaces and algebraic varieties 3. Subspaces 4. Partitions 5. Canonical forms for varieties and polarities 6. The line 7. Ovals 9. Arithmetic of arcs of degree two 10. Cubic curves 12. Arcs of higher degree 13. Blocking sets 14. Small planes 15.
Abstract: 1. Finite fields 2. Projective spaces and algebraic varieties 3. Subspaces 4. Partitions 5. Canonical forms for varieties and polarities 6. The line 7. First properties of the plane 8. Ovals 9. Arithmetic of arcs of degree two 10. Arcs in ovals 11. Cubic curves 12. Arcs of higher degree 13. Blocking sets 14. Small planes Appendix Notation References
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01 Jun 1986
TL;DR: An introduction to the theory of finite fields, with emphasis on those aspects that are relevant for applications, especially information theory, algebraic coding theory and cryptology and a chapter on applications within mathematics, such as finite geometries.
Abstract: The first part of this book presents an introduction to the theory of finite fields, with emphasis on those aspects that are relevant for applications. The second part is devoted to a discussion of the most important applications of finite fields especially information theory, algebraic coding theory and cryptology (including some very recent material that has never before appeared in book form). There is also a chapter on applications within mathematics, such as finite geometries. combinatorics. and pseudorandom sequences. Worked-out examples and list of exercises found throughout the book make it useful as a textbook.
1,819 citations
Book•
01 Jan 1992TL;DR: The second edition of a popular book on combinatorics as discussed by the authors is a comprehensive guide to the whole of the subject, dealing in a unified manner with, for example, graph theory, extremal problems, designs, colorings and codes.
Abstract: This is the second edition of a popular book on combinatorics, a subject dealing with ways of arranging and distributing objects, and which involves ideas from geometry, algebra and analysis. The breadth of the theory is matched by that of its applications, which include topics as diverse as codes, circuit design and algorithm complexity. It has thus become essential for workers in many scientific fields to have some familiarity with the subject. The authors have tried to be as comprehensive as possible, dealing in a unified manner with, for example, graph theory, extremal problems, designs, colorings and codes. The depth and breadth of the coverage make the book a unique guide to the whole of the subject. The book is ideal for courses on combinatorical mathematics at the advanced undergraduate or beginning graduate level. Working mathematicians and scientists will also find it a valuable introduction and reference.
1,678 citations
08 Apr 2009
888 citations
Cites background or methods or result from "Projective geometries over finite f..."
...Hence {L0, L1} and {L0, L1} are the reguli of an hyperbolic quadric Q+ of PG(3, q) [80]....
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...a (q + 2)-arc [80], of the projective plane PG(2, q), q = 2h, and let PG(2, q) = H be embedded as a plane in PG(3, q) = P ....
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...Then O′ is an oval with nucleus x [80]....
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...Since each plane of order s, s 6 8, is desarguesian [80], the result follows....
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...,Mq are contained in a three dimensional space P , and moreover {L0, L1} and {L0, L1} are the reguli of an hyperbolic quadric Q+ of P [80]....
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TL;DR: On etudie les relations entre les codes [n,k] lineaires a deux poids, les ensembles projectifs et certains graphes fortement reguliers as mentioned in this paper.
Abstract: On etudie les relations entre les codes [n,k] lineaires a deux poids, les ensembles (n,k,h 1 h 2 ) projectifs et certains graphes fortement reguliers
609 citations
Cites background from "Projective geometries over finite f..."
...Then O is a set of n = q+2 points, no three collinear, with the property that if L is a line then | L n O \ = 0 or 2. There are unique examples when q = 2 or q = 4 but many projectively different examples for large q (Hirschfeld [ 33, Chapter 8 ])....
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Book•
01 Jan 1992TL;DR: The standard geometric codes are presented, followed by a list of recommended designs and some examples of how these designs might be implemented in the real world.
Abstract: Algebraic coding theory has in recent years been increasingly applied to the study of combinatorial designs. This book gives an account of many of those applications together with a thorough general introduction to both design theory and coding theory - developing the relationship between the two areas. The first half of the book contains background material in design theory, including symmetric designs and designs from affine and projective geometry, and in coding theory, coverage of most of the important classes of linear codes. In particular, the authors provide a new treatment of the Reed-Muller and generalized Reed-Muller codes. The last three chapters treat the applications of coding theory to some important classes of designs, namely finite planes, Hadamard designs and Steiner systems, in particular the Witt systems. The book is aimed at mathematicians working in either coding theory or combinatorics - or related areas of algebra. The book is, however, designed to be used by non-specialists and can be used by those graduate students or computer scientists who may be working in these areas.
374 citations