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Chapter 1 Sphere Packings and Kissing Numbers&J.H.Conway and N.J.A.Sloane1

1．The Sphere Packing Problem1

1.1 Packing Ball Bearings1

1.2 Lattice Packings3

1.3 Nonlattice Packings7

1.4 n-Dimensional Packings8

1.5 Sphere Packing Problem—Summary of Results12

2．The Kissing Number Problem21

2.1 The Problem of the Thirteen Spheres21

2.2 Kissing Numbers in Other Dimensions21

2.3 Spherical Codes24

2.4 The Construction of Spherical Codes from Sphere Packings26

2.5 The Construction of Spherical Codes from Binary Codes26

2.6 Bounds on A(n,φ)27

Appendix:Planetary Perturbations29

Chapter 2 Coverings,Lattices and Quantizers&J.H.Conway and N.J.A.Sloane31

1．The Covering Problem31

1.1 Covering Space with Overlapping Spheres31

1.2 The Covering Radius and the Voronoi Cells33

1.3 Covering Problem—Summary of Results36

1.4 Computational Difficulties in Packings and Coverings40

2．Lattices,Quadratic Forms and Number Theory41

2.1 The Norm of a Vector41

2.2 Quadratic Forms Associated with a Lattice42

2.3 Theta Series and Connections with Number Theory44

2.4 Integral Lattices and Quadratic Forms47

2.5 Modular Forms50

2.6 Complex and Quaternionic Lattices52

3．Quantizers56

3.1 Quantization,Analog-to-Digital Conversion and Data Compression56

3.2 The Quantizer Problem59

3.3 Quantizer Problem—Summary of Results59

Chapter 3 Codes,Designs and Groups&J.H.Conway and N.J.A.Sloane63

1．The Channel Coding Problem63

1.1 The Sampling Theorem63

1.2 Shannon's Theorem66

1.3 Error Probability69

1.4 Lattice Codes for the Gaussian Channel71

2．Error-Correcting Codes75

2.1 The Error-Correcting Code Problem75

2.2 Further Definitions from Coding Theory77

2.3 Repetition,Even Weight and Other Simple Codes79

2.4 Cyclic Codes79

2.5 BCH and Reed-Solomon Codes81

2.6 Justesen Codes82

2.7 Reed-Muller Codes83

2.8 Quadratic Residue Codes84

2.9 Perfect Codes85

2.10 The Pless Double Circulant Codes86

2.11 Goppa Codes and Codes from Algebraic Curves87

2.12 Nonlinear Codes87

2.13 Hadamard Matrices87

3．t-Designs,Steiner Systems and Spherical t-Designs88

3.1 t-Designs and Steiner Systems88

3.2 Spherical t-Designs89

4．The Connections with Group Theory90

4.1 The Automorphism Group of a Lattice90

4.2 Constructing Lattices and Codes from Groups92

Chapter 4 Certain Important Lattices and Their Properties&J.H.Conway and N.J.A.Sloane94

1．Introduction94

2．Reflection Groups and Root Lattices95

3．Gluing Theory99

4．Notation;Theta Functions101

4.1 Jacobi Theta Functions102

5．The n-Dimensional Cubic Lattice Zn106

6．The n-Dimensional Lattices An and A* n108

6.1 The Lattice An108

6.2 The Hexagonal Lattice110

6.3 The Face-Centered Cubic Lattice112

6.4 The Tetrahedral or Diamond Packing113

6.5 The Hexagonal Close-Packing113

6.6 The Dual Lattice A* n115

6.7 The Body-Centered Cubic Lattice116

7．The n-Dimensional Lattices Dn and D* n117

7.1 The Lattice Dn117

7.2 The Four-Dimensional Lattice D4118

7.3 The Packing D* n119

7.4 The Dual Lattice D* n120

8．The Lattices E6,E7 and E8120

8.1 The 8-Dimensional Lattice E8120

8.2 The 7-Dimensional Lattices E7 and E* 7124

8.3 The 6-Dimensional Lattices E6 and E* 6125

9．The 12-Dimensional Coxeter-Todd Lattice K12127

10．The 16-Dimensional Barnes-Wall Lattice Λ16129

11．The 24-Dimensional Leech Lattice Λ24131

Chapter 5 Sphere Packing and Error-Correcting Codes&J.Leech and N.J.A.Sloane136

1．Introduction136

1.1 The Coordinate Array of a Point137

2．Construction A137

2.1 The Construction137

2.2 Center Density137

2.3 Kissing Numbers138

2.4 Dimensions 3 to 6138

2.5 Dimensions 7 and 8138

2.6 Dimensions 9 to 12139

2.7 Comparison of Lattice and Nonlattice Packings140

3．Construction B141

3.1 The Construction141

3.2 Center Density and Kissing Numbers141

3.3 Dimensions 8,9 and 12142

3.4 Dimensions 15 to 24142

4．Packings Built Up by Layers142

4.1 Packing by Layers142

4.2 Dimensions 4 to 7144

4.3 Dimensions 11 and 13 to 15144

4.4 Density Doubling and the Leech Lattice Λ24145

4.5 Cross Sections of Λ24145

5．Other Constructions from Codes146

5.1 A Code of Length 40146

5.2 A Lattice Packing in R40147

5.3 Cross Sections of Λ40148

5.4 Packings Based on Ternary Codes148

5.5 Packings Obtained from the Pless Codes148

5.6 Packings Obtained from Quadratic Residue Codes149

5.7 Density Doubling in R24 and R48149

6．Construction C150

6.1 The Construction150

6.2 Distance Between Centers150

6.3 Center Density150

6.4 Kissing Numbers151

6.5 Packings Obtained from Reed-Muller Codes151

6.6 Packings Obtained from BCH and Other Codes152

6.7 Density of BCH Packings153

6.8 Packings Obtained from Justesen Codes155

Chapter 6 Laminated Lattices&J.H.Conway and N.J.A.Sloane157

1．Introduction157

2．The Main Results163

3．Properties of Λ0 to Λ8168

4．Dimensions 9 to 16170

5．The Deep Holes in Λ16174

6．Dimensions 17 to 24176

7．Dimensions 25 to 48177

Appendix:The Best Integral Lattices Known179

Chapter 7 Further Connections Between Codes and Lattices&N.J.A.Sloane181

1．Introduction181

2 Construction A182

3．Self-Dual(or Type Ⅰ)Codes and Lattices185

4．Extremal Type Ⅰ Codes and Lattices189

5．Construction B191

6．Type Ⅱ Codes and Lattices191

7．Extremal Type Ⅱ Codes and Lattices193

8．Constructions A and B for Complex Lattices197

9．Self-Dual Nonbinary Codes and Complex Lattices202

10．Extremal Nonbinary Codes and Complex Lattices205

Chapter 8 Algebraic Constructions for Lattices&J.H.Conway and N.J.A.Sloane206

1．Introduction206

2 The Icosians and the Leech Lattice207

2.1 The Icosian Group207

2.2 The Icosian and Turyn-Type Constructions for the Leech Lattice210

3．A General Setting for Construction A,and Quebbemann's 64-Dimensional Lattice211

4．Lattices Over Z[e ni/4],and Quebbemann's 32-Dimensional Lattice215

5．McKay's 40-Dimensional Extremal Lattice221

6．Repeated Differences and Craig's Lattices222

7．Lattices from Algebraic Number Theory224

7.1 Introduction224

7.2 Lattices from the Trace Norm224

7.3 Examples from Cyclotomic Fields227

7.4 Lattices from Class Field Towers227

7.5 Unimodular Lattices with an Automorphism of Prime Order229

8．Constructions D and D'232

8.1 Construction D232

8.2 Examples233

8.3 Construction D'235

9．Construction E236

10．Examples of Construction E238

Chapter 9 Bounds for Codes and Sphere Packings&N.J.A.Sloane245

1．Introduction245

2．Zonal Spherical Functions249

2.1 The 2-Point-Homogeneous Spaces250

2.2 Representations of G252

2.3 Zonal Spherical Functions253

2.4 Positive-Definite Degenerate Kernels256

3．The Linear Programming Bounds257

3.1 Codes and Their Distance Distributions257

3.2 The Linear Programming Bounds258

3.3 Bounds for Error-Correcting Codes260

3.4 Bounds for Constant-Weight Codes263

3.5 Bounds for Spherical Codes and Sphere Packings263

4．Other Bounds265

Chapter 10 Three Lectures on Exceptional Groups&J.H.Conway267

1．First Lecture267

1.1 Some Exceptional Behavior of the Groups Ln(q)267

1.2 The Case p=3269

1.3 The Case p=5269

1.4 The Case p=7269

1.5 The Case p=11271

1.6 A Presentation for M12273

1.7 Janko's Group of Order 175560273

2．Second Lecture274

2.1 The Mathieu Group M24274

2.2 The Stabilizer of an Octad276

2.3 The Structure of the Golay Code ？24278

2.4 The Structure of P(Ω)/？24278

2.5 The Maximal Subgroups of M24279

2.6 The Structure of P(Ω)283

3．Third Lecture286

3.1 The Group Co0=·0 and Some ofits Subgroups286

3.2 The Geometry of the Leech Lattice286

3.3 The Group·0 and its Subgroup N287

3.4 Subgroups of·0290

3.5　The Higman-Sims and McLaughlin Groups292

3.6 The Group Co3=·3293

3.7 Involutions in·0294

3.8 Congruences for Theta Series294

3.9 A Connection Between·0 and Fischer's Group Fi24295

Appendix:On the Exceptional Simple Groups296

Chapter 11　The Golay Codes and the Mathieu Groups&J.H.Conway299

1．Introduction299

2．Definitions of the Hexacode300

3．Justification of a Hexacodeword302

4．Completing a Hexacodeword302

5．The Golay Code ？24 and the MOG303

6．Completing Octads from 5 of their Points305

7．The Maximal Subgroups of M24307

8．The Projective Subgroup L2(23)308

9．The Sextet Group 26:3·S6309

10．The Octad Group 24:A8311

11．The Triad Group and the Projective Plane of Order 4314

12．The Trio Group 26:(S3×L2(7))316

13．The Octern Group318

14．The Mathieu Group M23319

15．The Group M22:2319

16．The Group M12 the Tetracode and the MINIMOG320

17．Playing Cards and Other Games323

18．Further Constructions for M12327

Chapter 12 A Characterization of the Leech Lattice&J.H.Conway331

Chapter 13 Bounds on Kissing Numbers&A.M.Odlyzko and N.J.A.Sloane337

1．A General Upper Bound337

2．Numerical Results338

Chapter 14 Uniqueness of Certain Spherical Codes&E.Bannai and N.J.A.Sloane340

1．Introduction340

2．Uniqueness of the Code of Size 240 in Ω8342

3．Uniqueness of the Code of Size 56 in Ω7344

4．Uniqueness of the Code of Size 196560 in Ω24345

5．Uniqueness of the Code of Size 4600 in Ω23349

Chapter 15 On the Classification of Integral Quadratic Forms&J.H.Conway and N.J.A.Sloane352

1．Introduction352

2．Definitions354

2.1 Quadratic Forms354

2.2 Forms and Lattices:Integral Equivalence355

3．The Classification of Binary Quadratic Forms356

3.1 Cycles of Reduced Forms356

3.2 Definite Binary Forms357

3.3 Indefinite Binary Forms359

3.4 Composition of Binary Forms364

3.5 Genera and Spinor Genera for Binary Forms366

4．The p-Adic Numbers366

4.1 The p-Adic Numbers367

4.2 p-Adic Square Classes367

4.3 An Extended Jacobi-Legendre Symbol368

4.4 Diagonalization of Quadratic Forms369

5．Rational Invariants of Quadratic Forms370

5.1 Invariants and the Oddity Formula370

5.2 Existence of Rational Forms with Prescribed Invariants372

5.3　The Conventional Form of the Hasse-Minkowski Invariant373

6．The Invariance and Completeness of the Rational Invariants373

6.1 The p-Adic Invariants for Binary Forms373

6.2 The p-Adic Invariants for n-Ary Forms375

6.3 The Proof of Theorem 7377

7．The Genus and its Invariants378

7.1 p-Adic Invariants378

7.2 The p-Adic Symbol for a Form379

7.3 2-Adic Invariants380

7.4 The 2-Adic Symbol380

7.5 Equivalences Between Jordan Decompositions381

7.6 A Canonical 2-Adic Symbol382

7.7 Existence of Forms with Prescribed Invariants382

7.8 A Symbol for the Genus384

8．Classification of Forms of Small Determinant and of p-Elementary Forms385

8.1 Forms of Small Determinant385

8.2 p-Elementary Forms386

9．The Spinor Genus388

9.1 Introduction388

9.2 The Spinor Genus389

9.3 Identifying the Spinor Kernel390

9.4 Naming the Spinor Operators for the Genus of f390

9.5 Computing the Spinor Kernel from the p-Adic Symbols391

9.6 Tractable and Irrelevant Primes392

9.7 When is There Only One Class in the Genus?393

10．The Classification of Positive Definite Forms396

10.1 Minkowski Reduction396

10.2 The Kneser Gluing Method399

10.3 Positive Definite Forms of Determinant 2 and 3399

11．Computational Complexity402

Chapter 16 Enumeration of Unimodular Lattices&J.H.Conway and N.J.A.Sloane406

1．The Niemeier Lattices and the Leech Lattice406

2．The Mass Formulae for Lattices408

3．Verifications of Niemeier's List410

4．The Enumeration of Unimodular Lattices in Dimensions n≤23413

Chapter 17 The 24-Dimensional Odd Unimodular Lattices&R.E.Borcherds421

Chapter 18 Even Unimodular 24-Dimensional Lattices&B.B.Venkov429

1．Introduction429

2．Possible Configurations of Minimal Vectors430

3．On Lattices with Root Systems of Maximal Rank433

4．Construction of the Niemeier Lattices436

5．A Characterization of the Leech Lattice439

Chapter 19 Enumeration of Extremal Self-Dual Lattices&J.H.Conway,A.M.Odlyzko and N.J.A.Sloane441

1．Dimensions 1-16441

2．Dimensions 17-47441

3．Dimensions n≥48443

Chapter 20 Finding the Closest Lattice Point&J.H.Conway and N.J.A.Sloane445

1．Introduction445

2．The Lattices Zn,Dn and An446

3．Decoding Unions of Cosets448

4．"Soft Decision"Decoding for Binary Codes449

5．Decoding Lattices Obtained from Construction A450

6．Decoding E8450

Chapter 21 Voronoi Cells of Lattices and Quantization Errors&J.H.Conway and N.J.A.Sloane451

1．Introduction451

2．Second Moments of Polytopes453

2.A Dirichlet's Integral453

2.B Generalized Octahedron or Crosspolytope454

2.C The n-Sphere454

2.D n-Dimensional Simplices454

2.E Regular Simplex455

2.F Volume and Second Moment of a Polytope in Terms of its Faces455

2.G Truncated Octahedron456

2.H Second Moment of Regular Polytopes456

2.I Regular Polygons457

2.J Icosahedron and Dodecahedron457

2.K The Exceptional 4-Dimensional Polytopes457

3．Voronoi Cells and the Mean Squared Error of Lattice Quantizers458

3.A The Voronoi Cell of a Root Lattice458

3.B Voronoi Cell for An461

3.C Voronoi Cell for Dn(n≥4)464

3.D Voronoi Cells for E6,E7,E8464

3.E Voronoi Cell for D* n465

3.F Voronoi Cell for A* n474

3.G The Walls of the Voronoi Cell476

Chapter 22 A Bound for the Covering Radius of the Leech Lattice&S.P.Norton478

Chapter 23 The Covering Radius of the Leech Lattice&J.H.Conway,R.A.Parker and N.J.A.Sloane480

1．Introduction480

2．The Coxeter-Dynkin Diagram of a Hole482

3．Holes Whose Diagram Contains an An Subgraph486

4．Holes Whose Diagram Contains a Dn Subgraph497

5．Holes Whose Diagram Contains an En Subgraph504

Chapter 24 Twenty-Three Constructions for the Leech Lattice&J.H.Conway and N.J.A.Sloane508

1．The"Holy Constructions"508

2．The Environs of a Deep Hole512

Chapter 25　The Cellular Structure of the Leech Lattice&R.E.Borcherds,J.H.Conway and L.Queen515

1．Introduction515

2．Names for the Holes515

3．The Volume Formula516

4．The Enumeration of the Small Holes521

Chapter 26 Lorentzian Forms for the Leech Lattice&J.H.Conway and N.J.A.Sloane524

1．The Unimodular Lorentzian Lattices524

2．Lorentzian Constructions for the Leech Lattice525

Chapter 27 The Automorphism Group of the 26-Dimensional Even Unimodular Lorentzian Lattice&J.H.Conway529

1．Introduction529

2．The Main Theorem530

Chapter 28 Leech Roots and Vinberg Groups&J.H.Conway and N.J.A.Sloane534

1．The Leech Roots534

2．Enumeration of the Leech Roots543

3．The Lattices In·1 for n≤19549

4．Vinberg's Algorithm and the Initial Batches of Fundamental Roots549

5．The Later Batches of Fundamental Roots552

Chapter 29 The Monster Group and its 196884-Dimensional Space&J.H.Conway556

1．Introduction556

2．The Golay Code ？ and the Parker Loop p558

3．The Mathieu Group M24:the Standard Automorphisms of p558

4 The Golay Cocode ？* and the Diagonal Automorphisms558

5．The Group N of Triple Maps559

6．The Kernel K and the Homomorphism g→？559

7．The Structures of Various Subgroups of ？559

8．The Leech Lattice ∧24 and the Group ？560

9．Short Elements561

10．The Basic Representations of N？561

11．The Dictionary562

12．The Algebra563

13．The Definition of the Monster Group ？ and its Finiteness563

14．Identifying the Monster564

Appendix 1．Computing in p565

Appendix 2．A Construction for p565

Appendix 3．Some Relations in Q？566

Appendix 4．Constructing Representations for N？568

Appendix 5．Building the Group ？569

Chapter 30 A Monster Lie Algebra?&R.E.Borcherds,J.H.Conway,L.Queen and N.J.A.Sloane570

Bibliography574

Supplementary Bibliography642

Index681