Sphere packings, lattices, and groups / 3rd ed.
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作 者：J.H. Conway, N.J.A. Sloane ; with additional contributions by E. Bannai ... [et al.].
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ISBN：9787506292153
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简介
《球垛格点和群(第3版)》,继前两版之后，接着探讨“如何最有效地将大量等球放入n维的欧氏空间中？”这一核心问题。同时，作者仍在思考一些相关的问题，如：吻接数问题，覆盖问题，量子化问题以及格分类与二次型。与前两版相同的是，第三版也描述了以上这些问题与数学或自然科学中其他一些领域的联系，这些领域包括：码理论，数字通信，数论，群论，模拟数字转换以及数据压缩与n维晶体。值得特别注意的是，《球垛格点和群(第3版)》收录了一篇介绍本领域的最新的一些研究成果的报告，并补充了1988-1998年间出版的超过800项的参考书目，相信这些珍贵的资料一定能够引起读者特殊的兴趣。《球垛格点和群(第3版)》适用于数学专业的高年级本科生或研究生以及需要相关知识的科研人员。
目录
Preface to First Edition
Preface to Third Edition
List of Symbols
Chapter 1
Sphere Packings and Kissing Numbers
J.H. Conway and N.J.A. Sloane
1. The Sphere Packing Problem
1.1 Packing Ball Bearings
1.2 Lattice Packings
1.3 Nonlattice Packings
1.4 n-Dimensional Packings
1.5 Sphere Packing Problem-Summary of Results
2. The Kissing Number Problem
2.1 The Problem of the Thirteen Spheres
2.2 Kissing Numbers in Other Dimensions
2.3 Spherical Codes
2.4 The Construction of Spherical Codes from Sphere Packings
2.5 The Construction of Spherical Codes from Binary Codes
2.6 Bounds on A(n,)
Appendix: Planetary Perturbations
Chapter 2
Coverings, Lattices and Quantizers
J.H. Conway and N.J.A. Sloane
1. The Covering Problem
1.1 Covering Space with Overlapping Spheres
1.2 The Covering Radius and the Voronoi Cells
1.3 Covering Problem-Summary of Results
1.4 Computational Difficulties in Packings and Coverings
2. Lattices, Quadratic Forms and Number Theory
2.1 The Norm of a Vector
2.2 Quadratic Forms Associated with a Lattice
2.3 Theta Series and Connections with Number Theory
2.4 Integral Lattices and Quadratic Forms
2.5 Modular Forms
2.6 Complex and Quaternionic Lattices
3. Quantizers
3.1 Quantization, Analog-to-Digital Conversion and Data Compression
3.2 The Quantizer Problem
3.3 Quantizer Problem-Summary of Results
Chapter 3
Codes, Designs and Groups
J.H. Conway and N.J.A. Sloane
1. The Channel Coding Problem
1.1 The Sampling Theorem
1.2 Shannon's Theorem
1.3 Error Probability
1.4 Lattice Codes for the Gaussian Channel
2. Error-Correcting Codes
2.1 The Error-Correcting Code Problem
2.2 Further Definitions from Coding Theory
2.3 Repetition, Even Weight and Other Simple Codes
2.4 Cyclic Codes
2.5 BCH and Reed-Solomon Codes
2.6 Justesen Codes
2.7 Reed-Muller Codes
2.8 Quadratic Residue Codes
2.9 Perfect Codes
2.10 The Pless Double Circulant Codes
2.11 Goppa Codes and Codes from Algebraic Curves
2.12 Nonlinear Codes
2.13 Hadamard Matrices
3. t-Designs, Steiner Systems and Spherical t-Designs
3.1 t-Designs and Steiner Systems
3.2 Spherical t-Designs
4. The Connections with Group Theory
4.1 The Automorphism Group of a Lattice
4.2 Constructing Lattices and Codes from Groups
Chapter 4
Certain Important Lattices and Their Properties
J.H. Conway and N.J.A. Sloane
1. Introduction
2. Reflection Groups and Root Lattices
3. Gluing Theory
4. Notation; Theta Functions
4.1 Jacobi Theta Functions
5. The n-Dimensional Cubic Lattice Zn .
6. The n-Dimensional Lattices An and An*
6.1 The Lattice An.
6.2 The Hexagonal Lattice
6.3 The Face-Centered Cubic Lattice
6.4 The Tetrahedral or Diamond Packing
6.5 The Hexagonal Close-Packing
6.6 The Dual Lattice A*
6.7 The Body-Centered Cubic Lattice
7. The n-Dimensional Lattices Dn and Dn*
7.1 The Lattice Dn.
7.2 The Four-Dimensional Lattice D4 .
7.3 The Packing Dn
7.4The Dual Lattice Dn*
8. The Lattices E6, E7 and E8
8.1 The 8-Dimensional Lattice E8
8.2 The 7-Dimensional Lattices E7 and E7*
8.3 The 6-Dimensional Lattices E6and E6*
9. The 12-Dimensional Coxeter-Todd Lattice K12
10. The 16-Dimensional Barnes-Wall Lattice A16.
11. The 24-Dimensional Leech Lattice A24
Chapter 5
Sphere Packing and Error-Correcting Codes
J. Leech and N.J.A. Sloane
1. Introduction
1.1 The Coordinate Array of a Point
2. Construction A
2.1 The Construction
2.2 Center Density
2.3 Kissing Numbers
2.4 Dimensions 3 to 6
2.5 Dimensions 7 and 8
2.6 Dimensions 9 to 12
2.7 Comparison of Lattice and Nonlattice Packings
3. Construction B
3.1 The Construction
3.2 Center Density and Kissing Numbers
3.3 Dimensions 8, 9 and 12
3.4 Dimensions 15 to 24
4. Packings Built Up by Layers
4.1 Packing by Layers
4.2 Dimensions 4 to 7
4.3 Dimensions II and 13 to 15
4.4 Density Doubling and the Leech Lattice A,,
4.5 Cross Sections of A24,
5. Other Constructions from Codes
5.1 A Code of Length 40
5.2 A Lattice Packing in R40
5.3 Cross Sections of A40
5.4 Packings Based on Ternary Codes
5.5 Packings Obtained from the Pless Codes
5.6 Packings Obtained from Quadratic Residue Codes
5.7 Density Doubling in R24 and R48
6. Construction C
6.1 The Construction
6.2 Distance Between Centers
6.3 Center Density
6.4 Kissing Numbers
6.5 Packings Obtained from Reed-Muller Codes
6.6 Packings Obtained from BCH and Other Codes
6.7 Density of BCH Packings
6.8 Packings Obtained from Justesen Codes
Chapter 6
Laminated Lattices
J.H. Conway and N.J.A. Sloane
1. Introduction
2. The Main Results
3. Properties of A0 to A8
4. Dimensions 9 to 16
5. The Deep Holes in A16
6. Dimensions 17 to 24
7. Dimensions 25 to 48
Appendix: The Best Integral Lattices Known
Chapter 7
Further Connections Between Codes and Lattices
N.J.A. Sloane
1. Introduction
2. Construction A
3. Self-Dual (or Type I) Codes and Lattices
4. Extremal Type I Codes and Lattices
5. Construction B
6. Type Ⅱ Codes and Lattices
7. Extremal Type Ⅱ Codes and Lattices
8. Constructions A and B for Complex Lattices
9. Self-Dual Nonbinary Codes and Complex Lattices
10. Extremal Nonbinary Codes and Complex Lattices
Chapter 8
Algebraic Constructions for Lattices
J.H. Conway and N.J.A. Sloane
1. Introduction
2. The Icosians and the Leech Lattice
……
Chapter 9
Bounds for Codes and Sphere Packings
N.J.A. Sloane
Chapter 10
Three Lectures on Exceptional Groups
J.H. Conway
Chapter 11
The Golay Codes and the Mathieu Groups
J.H. Conway
Chapter 12
A Characterization of the Leech Lattice
J.H. Conway
Chapter 13
Bounds on Kissing Numbers
A.M. Odlyzko and N.J.A. Sloane
Chapter 14
Uniqueness of Certain Spherical Codes
E. Bannai and N.J.A. Sloane
Chapter 15
On the Classification of Integral Quadratic Forms
J.H. Conway and N.J.A. Sloane
Chapter 16
Enumeration of Unimodular Lattices
J.H. Conway and N.J.A. Sloane
Chapter 17
The 24-Dimensional Odd Unimodular Lattices
R.E. Borcherds
Chapter 18
Even Unimodular 24-Dimensional Lattices
B.B. Venkov
Chapter 19
Enumeration of Extremal Self-Dual Lattices
J.H. Conway, A.M. Odlyzko and N.J.A. Sloane
Chapter 20
Finding the Closest Lattice Point
J.H. Conway and N.J.A. Sloane
Chapter 21
Voronoi Cells of Lattices and Quantization Errors
J.H. Conway and N.J.A. SIoane
Chapter 22
A Bound for the Covering Radius of the Leech Lattice
S.P. Norton
Chapter 23
The Covering Radius of the Leech Lattice
J.H. Conway, R.A. Parker and N.J.A. Sloane
Chapter 24
Twenty-Three Constructions for the Leech Lattice
J.H. Conway and N.J.A. Sloane
Chapter 25
The Cellular Structure of the Leech Lattice
R.E. Borcherds, J.H. Conway and L. Queen
Chapter 26
Lorentzian Forms for the Leech Lattice
J.H. Conway and N.J.A. Sloane
Chapter 27
The Automorphism Group of the 26-Dimensional Even
Unimodular Lorentzian Lattice
J.H. Conway
Chapter 28
Leech Roots and Vinberg Groups
J.H. Conway and N.J.A. Sloane
Chapter 29
The Monster Group and its 196884-Dimensional Space
J.H. Conway
Chapter 30
A Monster Lie Algebra?
R.E. Borcherds, J.H. Conway, L. Queen and
N.J.A. Sloane
Bibliography
Supplementary Bibliography
Index
Preface to Third Edition
List of Symbols
Chapter 1
Sphere Packings and Kissing Numbers
J.H. Conway and N.J.A. Sloane
1. The Sphere Packing Problem
1.1 Packing Ball Bearings
1.2 Lattice Packings
1.3 Nonlattice Packings
1.4 n-Dimensional Packings
1.5 Sphere Packing Problem-Summary of Results
2. The Kissing Number Problem
2.1 The Problem of the Thirteen Spheres
2.2 Kissing Numbers in Other Dimensions
2.3 Spherical Codes
2.4 The Construction of Spherical Codes from Sphere Packings
2.5 The Construction of Spherical Codes from Binary Codes
2.6 Bounds on A(n,)
Appendix: Planetary Perturbations
Chapter 2
Coverings, Lattices and Quantizers
J.H. Conway and N.J.A. Sloane
1. The Covering Problem
1.1 Covering Space with Overlapping Spheres
1.2 The Covering Radius and the Voronoi Cells
1.3 Covering Problem-Summary of Results
1.4 Computational Difficulties in Packings and Coverings
2. Lattices, Quadratic Forms and Number Theory
2.1 The Norm of a Vector
2.2 Quadratic Forms Associated with a Lattice
2.3 Theta Series and Connections with Number Theory
2.4 Integral Lattices and Quadratic Forms
2.5 Modular Forms
2.6 Complex and Quaternionic Lattices
3. Quantizers
3.1 Quantization, Analog-to-Digital Conversion and Data Compression
3.2 The Quantizer Problem
3.3 Quantizer Problem-Summary of Results
Chapter 3
Codes, Designs and Groups
J.H. Conway and N.J.A. Sloane
1. The Channel Coding Problem
1.1 The Sampling Theorem
1.2 Shannon's Theorem
1.3 Error Probability
1.4 Lattice Codes for the Gaussian Channel
2. Error-Correcting Codes
2.1 The Error-Correcting Code Problem
2.2 Further Definitions from Coding Theory
2.3 Repetition, Even Weight and Other Simple Codes
2.4 Cyclic Codes
2.5 BCH and Reed-Solomon Codes
2.6 Justesen Codes
2.7 Reed-Muller Codes
2.8 Quadratic Residue Codes
2.9 Perfect Codes
2.10 The Pless Double Circulant Codes
2.11 Goppa Codes and Codes from Algebraic Curves
2.12 Nonlinear Codes
2.13 Hadamard Matrices
3. t-Designs, Steiner Systems and Spherical t-Designs
3.1 t-Designs and Steiner Systems
3.2 Spherical t-Designs
4. The Connections with Group Theory
4.1 The Automorphism Group of a Lattice
4.2 Constructing Lattices and Codes from Groups
Chapter 4
Certain Important Lattices and Their Properties
J.H. Conway and N.J.A. Sloane
1. Introduction
2. Reflection Groups and Root Lattices
3. Gluing Theory
4. Notation; Theta Functions
4.1 Jacobi Theta Functions
5. The n-Dimensional Cubic Lattice Zn .
6. The n-Dimensional Lattices An and An*
6.1 The Lattice An.
6.2 The Hexagonal Lattice
6.3 The Face-Centered Cubic Lattice
6.4 The Tetrahedral or Diamond Packing
6.5 The Hexagonal Close-Packing
6.6 The Dual Lattice A*
6.7 The Body-Centered Cubic Lattice
7. The n-Dimensional Lattices Dn and Dn*
7.1 The Lattice Dn.
7.2 The Four-Dimensional Lattice D4 .
7.3 The Packing Dn
7.4The Dual Lattice Dn*
8. The Lattices E6, E7 and E8
8.1 The 8-Dimensional Lattice E8
8.2 The 7-Dimensional Lattices E7 and E7*
8.3 The 6-Dimensional Lattices E6and E6*
9. The 12-Dimensional Coxeter-Todd Lattice K12
10. The 16-Dimensional Barnes-Wall Lattice A16.
11. The 24-Dimensional Leech Lattice A24
Chapter 5
Sphere Packing and Error-Correcting Codes
J. Leech and N.J.A. Sloane
1. Introduction
1.1 The Coordinate Array of a Point
2. Construction A
2.1 The Construction
2.2 Center Density
2.3 Kissing Numbers
2.4 Dimensions 3 to 6
2.5 Dimensions 7 and 8
2.6 Dimensions 9 to 12
2.7 Comparison of Lattice and Nonlattice Packings
3. Construction B
3.1 The Construction
3.2 Center Density and Kissing Numbers
3.3 Dimensions 8, 9 and 12
3.4 Dimensions 15 to 24
4. Packings Built Up by Layers
4.1 Packing by Layers
4.2 Dimensions 4 to 7
4.3 Dimensions II and 13 to 15
4.4 Density Doubling and the Leech Lattice A,,
4.5 Cross Sections of A24,
5. Other Constructions from Codes
5.1 A Code of Length 40
5.2 A Lattice Packing in R40
5.3 Cross Sections of A40
5.4 Packings Based on Ternary Codes
5.5 Packings Obtained from the Pless Codes
5.6 Packings Obtained from Quadratic Residue Codes
5.7 Density Doubling in R24 and R48
6. Construction C
6.1 The Construction
6.2 Distance Between Centers
6.3 Center Density
6.4 Kissing Numbers
6.5 Packings Obtained from Reed-Muller Codes
6.6 Packings Obtained from BCH and Other Codes
6.7 Density of BCH Packings
6.8 Packings Obtained from Justesen Codes
Chapter 6
Laminated Lattices
J.H. Conway and N.J.A. Sloane
1. Introduction
2. The Main Results
3. Properties of A0 to A8
4. Dimensions 9 to 16
5. The Deep Holes in A16
6. Dimensions 17 to 24
7. Dimensions 25 to 48
Appendix: The Best Integral Lattices Known
Chapter 7
Further Connections Between Codes and Lattices
N.J.A. Sloane
1. Introduction
2. Construction A
3. Self-Dual (or Type I) Codes and Lattices
4. Extremal Type I Codes and Lattices
5. Construction B
6. Type Ⅱ Codes and Lattices
7. Extremal Type Ⅱ Codes and Lattices
8. Constructions A and B for Complex Lattices
9. Self-Dual Nonbinary Codes and Complex Lattices
10. Extremal Nonbinary Codes and Complex Lattices
Chapter 8
Algebraic Constructions for Lattices
J.H. Conway and N.J.A. Sloane
1. Introduction
2. The Icosians and the Leech Lattice
……
Chapter 9
Bounds for Codes and Sphere Packings
N.J.A. Sloane
Chapter 10
Three Lectures on Exceptional Groups
J.H. Conway
Chapter 11
The Golay Codes and the Mathieu Groups
J.H. Conway
Chapter 12
A Characterization of the Leech Lattice
J.H. Conway
Chapter 13
Bounds on Kissing Numbers
A.M. Odlyzko and N.J.A. Sloane
Chapter 14
Uniqueness of Certain Spherical Codes
E. Bannai and N.J.A. Sloane
Chapter 15
On the Classification of Integral Quadratic Forms
J.H. Conway and N.J.A. Sloane
Chapter 16
Enumeration of Unimodular Lattices
J.H. Conway and N.J.A. Sloane
Chapter 17
The 24-Dimensional Odd Unimodular Lattices
R.E. Borcherds
Chapter 18
Even Unimodular 24-Dimensional Lattices
B.B. Venkov
Chapter 19
Enumeration of Extremal Self-Dual Lattices
J.H. Conway, A.M. Odlyzko and N.J.A. Sloane
Chapter 20
Finding the Closest Lattice Point
J.H. Conway and N.J.A. Sloane
Chapter 21
Voronoi Cells of Lattices and Quantization Errors
J.H. Conway and N.J.A. SIoane
Chapter 22
A Bound for the Covering Radius of the Leech Lattice
S.P. Norton
Chapter 23
The Covering Radius of the Leech Lattice
J.H. Conway, R.A. Parker and N.J.A. Sloane
Chapter 24
Twenty-Three Constructions for the Leech Lattice
J.H. Conway and N.J.A. Sloane
Chapter 25
The Cellular Structure of the Leech Lattice
R.E. Borcherds, J.H. Conway and L. Queen
Chapter 26
Lorentzian Forms for the Leech Lattice
J.H. Conway and N.J.A. Sloane
Chapter 27
The Automorphism Group of the 26-Dimensional Even
Unimodular Lorentzian Lattice
J.H. Conway
Chapter 28
Leech Roots and Vinberg Groups
J.H. Conway and N.J.A. Sloane
Chapter 29
The Monster Group and its 196884-Dimensional Space
J.H. Conway
Chapter 30
A Monster Lie Algebra?
R.E. Borcherds, J.H. Conway, L. Queen and
N.J.A. Sloane
Bibliography
Supplementary Bibliography
Index
Sphere packings, lattices, and groups / 3rd ed.
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